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STOCHASTIC MODELING OF ELECTRICITY MARKETS
A THESIS SUBMITTED TOTHE GRADUATE SCHOOL OF APPLIED MATHEMATICS
OFMIDDLE EAST TECHNICAL UNIVERSITY
BY
IREM TALASLI
IN PARTIAL FULFILLMENT OF THE REQUIREMENTSFOR
THE DEGREE OF PHILOSOPHY OF DOCTORATEIN
FINANCIAL MATHEMATICS
JANUARY 2012
Approval of the thesis:
STOCHASTIC MODELING OF ELECTRICITY MARKETS
submitted byIREM TALASLI in partial fulfillment of the requirements for the degree ofPhilosophy of Doctorate in Department of Financial Mathematics, Middle East Techni-cal University by,
Prof. Dr. Ersan AkyıldızDirector, Graduate School ofApplied Mathematics
Assoc. Prof. Dr.Omur UgurHead of Department,Financial Mathematics
Assoc. Prof. Dr. Azize HayfaviSupervisor,Department of Financial Mathematics
Examining Committee Members:
Prof. Dr. Gerhard Wilhelm WeberInstitute of Applied Mathematics, METU
Prof. Dr. Gul ErgunDepartment of Statistics, Hacettepe University
Assoc. Prof. Dr. Azize HayfaviInstitute of Applied Mathematics, METU
Assoc. Prof. Dr.Omur UgurInstitute of Applied Mathematics, METU
Assist. Prof. Dr. Yeliz Yolcu OkurInstitute of Applied Mathematics, METU
Date:
∗ Write the country name for the foreign committee member.
I hereby declare that all information in this document has been obtained and presentedin accordance with academic rules and ethical conduct. I also declare that, as requiredby these rules and conduct, I have fully cited and referenced all material and results thatare not original to this work.
Name, Last Name: IREM TALASLI
Signature :
iii
ABSTRACT
STOCHASTIC MODELING OF ELECTRICITY MARKETS
Talaslı,Irem
Ph.D., Department of Financial Mathematics
Supervisor : Assoc. Prof. Dr. Azize Hayfavi
January 2012, 88 pages
Day-ahead spot electricity markets are the most transparent spot marketswhere one can find
integrated supply and demand curves of the market players for each settlement period. Since
it is an indicator for the market players and regulators, in this thesis we modelthe spot elec-
tricity prices. Logarithmic daily average spot electricity prices are modeled asa summation
of a deterministic function and multi-factor stochastic process. Randomness in the spot prices
is assumed to be governed by three jump processes and a Brownian motion where two of the
jump processes are mean reverting. While the Brownian motion captures daily regular price
movements, the pure jump process models price shocks which have long term effects and
two Ornstein Uhlenbeck type jump processes with different mean reversion speeds capturing
the price shocks that affect the price level for relatively shorter time periods. After remov-
ing the seasonality which is modeled as a deterministic function from price observations, an
iterative threshold function is used to filter the jumps. The threshold function isconstructed
on volatility estimation generated by a GARCH(1,1) model. Not only the jumps but also the
mean reverting returns following the jumps are filtered. Both of the filtered jump processes
and residual Brownian components are estimated separately. The model is applied to Aus-
trian, Italian, Spanish and Turkish electricity markets data and it is found thatthe weekly
iv
forecasts, which are generated by the estimated parameters, turn out to beable to capture the
characteristics of the observations.
After examining the future contracts written on electricity, we also suggest a decision tech-
nique which is built on risk premium theory. With the help of this methodology derivative
market players can decide on taking whether a long or a short position fora given contract.
After testing our technique, we conclude that the decision rule is promising but needs more
empirical research.
Keywords: Electricity spot price, stochastic multi-factor model, jump process, GARCH (1,1),
risk premium
v
OZ
ELEKTRIK PIYASALARININ STOKASTIK MODELLEMESI
Talaslı,Irem
Doktora, Finansal Matematik Bolumu
Tez Yoneticisi : Doc. Dr. Azize Hayfavi
Ocak 2012, 88 sayfa
Gun oncesi elektrik piyasaları, piyasa oyuncularının her uzlasma donemine ait toplam arz
ve talep egrilerini iceren en seffaf spot elektrik piyasasıdır. Bu tezde hem piyasa oyuncuları
hem de duzenleme kurumları acısından gosterge niteligi tasıyan gun oncesi elektrik fiyat-
ları modellenmektedir. Logaritması alınan gunluk ortalama spot elektrik fiyatları determin-
istik bir fonsiyon ve cok faktorlu stokastik surecler ile modellenmistir. Spot fiyalardaki ras-
sallık iki tanesi ortalamaya donen olmakuzereuc adet sıcrama sureci ve bir Brown hareketi
tarafından yonetilmektedir. Brown hareketi gunluk olagan fiyat hareketlerini yakalarken,
sıcrama sureclerinden ilki uzun donemde etkili olan fiyat soklarının, farklı ortalamaya donus
hızlarına sahip olan iki tane Ornstein Uhlenbeck tipi sıcrama sureci ise fiyat seviyesiuzerinde
gorece daha kısa sure etkili olan fiyat soklarının acıklanmasında kullanılmaktadır. Determin-
istik bir fonksiyon ile modellenen mevsimsellik etkisinin gozlenen fiyatlardan ayrıstırlmasından
sonra, iteratif esik fonksiyonu kullanılarak fiyatlardaki sıcramalar filtrelenmektedir. Soz konusu
esik fonksiyonunda GARCH(1,1) kullanılarak heasplanan oynaklık tahminleri kullanılmaktadır.
Sadece sıcramalar degil, onları takip eden ortalamaya donus getirileri de filtre yardımı ile
ayrılmaktadır. Filtrelenen sıcrama surecleri ve bu surecten geriye kalan Brown hareketine ait
parametreler ayrı ayrı hesaplanmaktadır. Model Avusturya,Italya, Ispanya ve Turkiye’deki
vi
elektrik piyasa verilerine uygulanmıs ve elde edilen parametereler kullanılarak yapılan haf-
talık fiyat tahminlerinin fiyat gozlemlerine yakınsadıgı gorulmustur.
Elektrik uzerine yazılan vadeli sozlesmeler incelenerek, risk primi teorisine dayalı bir yontem
onerilmistir. Bu yontem piyasa oyuncularına, herhangi bir vadeli elektirk sozlesmesinde kısa
ya da uzun pozisyon alınması dogrultusunda bilgi vermektedir. Yapılan testler sonucunda
soz konusu yontemin uygulabilir oldugu ancak daha fazla ampirik calısmayla desteklenmesi
gerektigi sonucuna varılmıstır.
Anahtar Kelimeler: Spot elektrik fiyatı, stokastik faktor modelleri, sıcrama surecleri, GARCH
(1,1), risk primi
vii
In the memory of Prof. Dr. Hayri Korezlioglu
viii
ACKNOWLEDGMENTS
I would like to express my sincere thanks to everyone who contributed to my research and
have helped me in the formation of this thesis.
I would like to express my gratitude to my advisor Assoc. Prof. Dr. Azize Hayfavi for her
guidance and encouragement throughout the whole process.
I am grateful to Assoc. Prof. Dr.Omur Ugur, Assist. Prof. Dr. Yeliz Yolcu Okur, Prof. Dr.
Gerhard Wilhelm Weber and Assist. Prof. Dr. Kasırga Yıldırak for their guidance.
I would like to express my special thanks to my parents, Azra and Aykut Yıldırım and to my
husband Anıl Talaslı for their endless support and patience.
I am thankful to Zehra Eksi Altay, Aslı Kunt and all of my friends in IAM for their valuable
help.
Finally I would like to thank all of my colleagues in the Central Bank of Turkey,for the pa-
tience and tolerance they have shown to me. I would like to express my speacial thanks to Dr.
Emrah Eksi, Dr. Esref Mehmet Goksu, Bengi Deliorman,Ozlem Kumbaracı, Metin Kavut,
Ozgur Ergul, KevserOzturk, Nihan Guvendik, Fusun Esen, Arzu Agcaer, Bulent Yaman,
Bulent Usta and Suat Aydın.
ix
TABLE OF CONTENTS
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv
OZ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi
DEDICATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viii
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
TABLE OF CONTENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
CHAPTERS
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 SPOT ELECTRICITY MARKETS . . . . . . . . . . . . . . . . . . . . . . . 4
2.1 Characteristics of Electricity as a Commodity . . . . . . . . . . . . 6
2.2 The Motivation in Modeling Spot Electricity Prices . . . . . . . . . 8
2.3 Spot Electricity Models in Literature . . . . . . . . . . . . . . . . . 10
3 THE SPOT PRICE MODEL . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1 Deterministic Part . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.2 Jump Detection with Threshold Method . . . . . . . . . . . . . . . 23
3.3 Determining The Threshold . . . . . . . . . . . . . . . . . . . . . . 27
3.4 Decomposition of the Stochastic Processes . . . . . . . . . . . . . . 31
4 ESTIMATION OF SPOT ELECTRICITY PRICE PROCESS AND FORE-CATING SPOT ELECTRICITY PRICES . . . . . . . . . . . . . . . . . . . 37
4.1 Model Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2 Forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
5 ELECTRICITY FUTURES MARKETS . . . . . . . . . . . . . . . . . . . . 48
5.1 Stochastic Futures Pricing Models in Literature . . . . . . . . . . . 50
x
5.1.1 Spot price approach . . . . . . . . . . . . . . . . . . . . . 51
5.1.2 Instantaneous delivery models . . . . . . . . . . . . . . . 53
5.1.3 Direct modeling of the observed future contracts . . . . . 56
5.2 Risk Premium Approach . . . . . . . . . . . . . . . . . . . . . . . . 58
6 CONCLUSION AND OUTLOOK . . . . . . . . . . . . . . . . . . . . . . . 64
APPENDICES
A Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.2 Heath Jarrow Merton Framework for the Stochastic Modeling of In-terest Rate Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . 66
A.3 Risk Premium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
B Matlab Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
xi
LIST OF TABLES
TABLES
Table 4.1 Day of the week parameters . . . . . . . . . . . . . . . . . . . . . . . . . .40
Table 4.2 Deterministic function parameters . . . . . . . . . . . . . . . . . . . . . . 40
Table 4.3 Number of jumps filtered by the iterative GARCH(1,1) volatility threshold
r(h) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Table 4.4 Jump intensities=number of filtered jumps/total number of returns . . . . . 43
Table 4.5 Discrete time mean reversion rates . . . . . . . . . . . . . . . . . . . . . . 43
Table 4.6 Jump distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Table 5.1 Suggested future contract position according to the risk premium .. . . . . 63
xii
LIST OF FIGURES
FIGURES
Figure 2.1 Demand and supply curves from Italian spot electricity market . .. . . . . 7
Figure 2.2 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .. . 7
Figure 4.1 Daily average price series . . . . . . . . . . . . . . . . . . . . . . . . .. 38
Figure 4.2 Estimated weekly cycle for Italian electricity market . . . . . . . . . . . .39
Figure 4.3 Decomposition of deterministic and stochastic components . . . . . . . .. 41
Figure 4.4 Separated jumps and mean reverting returns . . . . . . . . . . . . . .. . 42
Figure 4.5 Autocorrelation functions of the original return series and the filtered re-
turns after the jump series are extracted . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 4.6 QQ-plots of the original return series and the filtered returns . .. . . . . . 45
Figure 4.7 Observed and forecasted price series starting from 2011 . .. . . . . . . . 46
Figure 4.8 Observed and forecasted price series through the forecast period . . . . . 46
Figure 4.9 MWE of each week in the forecast period and their average . .. . . . . . 47
Figure 5.1 TURDEX-December 2011 contract . . . . . . . . . . . . . . . . . . .. . 61
Figure 5.2 Optional caption for list of figures . . . . . . . . . . . . . . . . . . . .. . 62
Figure 5.3 Suggested future positions’ pay offs . . . . . . . . . . . . . . . . . . . . . 63
xiii
CHAPTER 1
INTRODUCTION
In the past two decades, electricity industries in many countries which were initially de-
signed as vertically integrated national or state dominated monopolies have been experienc-
ing a deregulation process. Throughout this process, generation, transmission, distribution
and marketing activities have been seperated and opened to competition wherever it is pos-
sible and profitable. The rise in the number of market players together with thecompetition
and development of relatively more liberal electricity markets caused spot prices to be de-
termined by supply and demand. This development increased the importance of accurate
electricity price forecasting and therefore electricity price modeling. As a commodity, elec-
tricity has different characteristics than other commodities as well as financial assets dueto
its non-storability, its demand inelasticity and significant seasonality of its consumption and
production. Electricity consumption is mainly influenced by industrial production, business
activities and weather conditions, therefore electricity prices show intra daily, weekly and
annual seasonal behaviour. Electricity prices are also mean reverting processes with high
volatility and sharp price spikes. Non-storability of electrical energy and necessity of con-
tinuous supply and demand balance in the transmission mechanism cause direct reflection of
supply and demand shocks on electricity prices. As soon as these extreme conditions dis-
appear, their effects on price levels also fade away. Additional to bilateral electricity trade
contracts, whose conditions are predetermined, a significant amount of electricity trade takes
place in the spot markets. The spot electricity markets are completed with electricity deriva-
tives trading taking place both in over the counter markets and derivative exchanges. Because
of the non-storability of electricity, it is not possible to use main stream derivatives valuation
models based on no arbitrage theory. However, due to their high informative content about
market players’ expectations or hedging behaviour of the agents, derivative accurate modeling
1
of electricity derivatives is a necessity.
We model the daily series of equally weighted average of system marginal prices for each
day’s balancing intervals (hour/ half an hour) formed in the day-ahead markets. Due to mean
reversion property of electricity prices, modeling with Ornstein Uhlenbeck type processes are
very common in literature. The need for a better fit of model forecasts with observations
leads to more complicated models with latent variables and multiple regimes. However, with
the increasing number of parameters, the estimation process also gets complicated. Filtering
techniques are one of the widely used methods in parameter estimation of the jump processes.
In our spot electricity price model, three jump processes without predetermined jump size dis-
tributions and a Brownian motion are combined with a deterministic seasonality term. While
the Brownian motion captures the daily regular price movements, the pure jump process mod-
els the price shocks which have long term effects and two Ornstein Uhlenbeck type processes
with different mean reversion speeds that capture the price shocks that have short term effects.
One of the mean reverting processes is assumed to model price shocks thatrevert back in the
next observation (price spikes) and the other is assumed to model price shocks that take a few
days to fade away (semi-spikes). An iterative threshold derived by using estimated volatility
with GARCH(1,1) is used to filter the price jumps. We construct an algorithm forstep by step
parameter estimation of a multi-factor price model.
One of the main goals of this thesis is to propose spot and future contract price models which
can be used in recently established liberal Turkish electricity spot market and electricity future
contracts that are traded in national derivatives exchange. However, since we do not prefer to
be restrained by the small sample size, we test our model with relatively more mature markets’
spot electricity data. We apply our spot price model to four different countries; Austria, Italy,
Spain and Turkey all of which heavily depend on thermal sources for electricity generation.
The hydro sources in these countries generally follow thermal sources interms of electricity
generation. In all of the four countries, a higher proportion of the generated electricity is
consumed by the industry. Austria, Spain and Italy have mature markets relative to Turkish
day-ahead spot electricity market which has been taken into full operationin December 2009
and is still evolving through its final market design. For all of the examined countries, the
separation results are found to be in accordance with our initial expectations.
In the analysis of the electricity futures pricing, we summarize three main approaches used
2
for electricity forward and future contracts modeling in the literature. We cannot propose a
contract price model due to data shortage. Instead of proposing a future contracts price model,
we offer a decision technique where the given contract prices are used. With this technique
which is built on the risk premium theory, derivative market players can decide taking whether
a long or a short position. After testing our technique, we conclude that thedecision rule is
promising but needs more empirical research.
In this context, spot electricity price processes are discussed in detail in Chapter 2. Our
multi-factor spot electricity price model is summarized in Chapter 3. Chapter is devoted to
the estimation of model parameters and Monte Carlo simulations. Chapter 5 is dedicated to
electricity futures.
3
CHAPTER 2
SPOT ELECTRICITY MARKETS
Spot electricity, referring to electrical power produced for current consumption, is provided
by a transformation industry which can be split into three processes.
• Electricity is produced by generators, who burn fuels such as coal, natural gas or nu-
clear fuel in power plants or use the gravitational energy of water or the wind force.
Generated energy is then injected into a high voltage network.
• Network operator (an independent system operator) is responsible for the transmission
of electricity and protection of the global balance between electricity given into and
taken out of the transmission system in order to prevent a possible collapse. It operates
a software system that allows the agents to exchange electricity on the high voltage
network, which is the natural place for wholesale trading.
• Marketing companies and distributors get the electricity from the high voltage network,
cascade it down to network distributions with lower voltage and sell it to industrial or
residential consumers and handle the metering and billing.
A few decades ago, in most of the countries worldwide, electricity sector was a vertically
integrated industry, where generation transmission and distribution is done by a single en-
tity. In this setting, prices were determined by regulators to reflect the cost of generation,
transmission and distribution and these prices were used to change in a deterministic manner.
However, over the last twenty years, electricity markets in many countries are experiencing
a deregulation process, aiming to introduce competition in generation, supply activities and
distribution. The first step in this deregulation process is unboundling the activities conducted
by the national monopolies. Unboundling the vertically integrated utilities means identifying
4
and separating the different tasks attached to a single entity in the traditional organization.
Then these tasks are opened to competition wherever it is possible and profitable. Contrary
to generation and distribution, which are appropriate to adapt a competitive market structure,
transmission is generally classified as a natural monopoly that should be operated by a single
system operator. Managing the grid system requires continuously keeping the entire transmis-
sion system at equilibrium. Since it is easier to manage the whole system from a single control
point, transmission has generally left out from the liberalization and privatization processes.
The upsurge in the number of market players together with the competition and development
of relatively more liberal electricity markets caused electricity to become a commodity whose
price is determined by supply and demand. In this market design, long-term bilateral con-
tracts between generators and wholesales (and retailers) dominate the mostof the electricity
trading. However, if approximately 80% of the trade takes place accordingto the prede-
termined conditions of these bilateral contracts, the remaining 20% takes placein the spot
electricity markets. The spot electricity market refers to the electricity trading that takes place
in a day-ahead market, which is managed by the system operator. For their residual genera-
tion capacity that is not bounded by bilateral agreements or for their extra electricity demand,
market players send their bids to the system operator in terms of prices and quantities for each
settlement period (generally settlement periods are hours or half hours) of the following day.
The system operator collect these bids, ranks them by merit order from the least expensive to
the next least expensive and so forth, then builds the supply and demandfunctions. Two of
the possible designs that the system operator can manage are given below.
• It is only the suppliers that make bids and the system operator is responsiblefor comput-
ing the expected demand for each settlement period of the following day; intersecting
this demand with the supply function provides the system marginal price.
• It is both buyers and sellers who make bids to the pool and then the system operator has
to build an analogous demand function, which is a quasivertical line since electricity
demand is fairly inelastic to price changes. The marginal price is again defined by the
intersection of the curves.
Contrary to bilateral contracts whose conditions are not publicly announced, system clearing
prices published every day by the operator. The latter is assumed to be a good indicator
5
for the spot electricity markets. On the other hand, the ultimate role of a spot market is
to ensure that total generation meets total demand. Although most of the energy trading is
scheduled in advance (from years to day ahead), some imbalances unavoidably occur in real
time because of various generation, load, and transmission factors. Generation factors include
plant outages, generators not following their schedules accurately, theuse of some generation
to provide ancillary services, and the intermittent nature of some generation sources (like
wind). Load factors include sudden changes in weather forecasting errors, and intrahour load
changes. Transmission factors include local and regional congestion, unscheduled flows, and
forced outages. All these factors, separetly and in combination, requirethe system operator
to have access to generation output so that it can move up or down from interval to interval.
This real time balancing is provided by the reserve capacity mechanism where market players
are obliged to hold a reserve capacity, which can be taken into operation in afew minutes if
the system operator needs it.
2.1 Characteristics of Electricity as a Commodity
As a commodity, electricity has different characteristics than other commodities and financial
assets due to its non-storability, its demand inelasticity and significant seasonality effect in
its consumption and production. Unlike the financial assets traded for investment purposes,
electricity is traded in order to be consumed. This close link with the real economy and daily
life causes electricity price to exhibit a different behaviour than those of the financial assets.
As a secondary energy source, created by the conversion of the other energy resources, elec-
tricity is difficult and expensive to store or transmit between different regions. Therefore, the
spot price of the electricity is set by the short term supply demand equilibrium,and as already
stressed, this equilibrium should be maintained at all times for the safety and thecontinuity
of the whole regional transmission system. Theoretical supply and demand curves of an hour,
in a day-ahead market is given in Figure 2.1. This figure is taken from the Italian system
operator GME’s web site [46] and it shows the supply and demand curvesof 12th hour of
30 June 2011, which are constructed according to market players’ bids. In Figure 2.2(a) the
hourly equilibrium prices for a random week are illustrated. And a longer series of daily
prices, computed from the daily averages of hourly prices, are shown inFigure 2.2(b).
6
Figure 2.1: Demand and supply curves from Italian spot electricity market
These representative series can help us identify some of the outstanding features of spot prices
for electricity.
(a) Hourly electricity prices in Turkish spot electricitymarket between 1-15 March 2010
(b) Daily electricity prices in Turkish spot electricitymarket between 1 Dec 09 - 30 Sep 2010
Figure 2.2: Spot electricity prices
1. Seasonal Behavior:Electricity consumption is mainly influenced by economic cycles,
industrial production, business activities and weather conditions. Therefore electricity
prices show intra daily, weekly and annual seasonal behaviour. In accordance with the
literature, we assume that seasonal behaviour is governed by a deterministic function.
Since we will work with daily average prices, only weekly and yearly seasonality will
7
be considered.
2. Mean Reversion:Contrary to stock prices which can be evolved freely in any direction,
electricity prices are known for their anti-persistent nature against supply and demand
shocks. Electricity prices generally gravitate around the cost of production. Abnormal
market conditions may lead to price spreads in the short run, but in the long run supply
will be adjusted and prices will return to the level dictated by the cost of production.
Moreover, one of the main factors determining electricity consumption is the temper-
ature which is also a mean reverting process. As a result, spot electricity prices show
strong mean reverting behaviour.
3. High Volatility and Sharp Spikes: Non-storability of electrical energy and necessity
of continuous supply and demand balance in the transmission mechanism cause direct
reflection of supply and demand shocks on electricity prices. As it can be seen from
Figure 2.1 electricity demand is highly inelastic. The characteristic of the supplystack
can also contribute to the price volatility. For low levels of demand, generatorssupply
electricity by using base load units with low marginal costs; as higher quantities needed,
generation plants with high marginal costs enter into the system. Therefore higher
demand levels may lead to jumps in the prices observed. Moreover, in case ofa plant
failure, high cost generation plants have to supply energy to the system. However, when
the conditions causing the price shocks disappear, prices revert backto their long term
equilibrium levels.
2.2 The Motivation in Modeling Spot Electricity Prices
Market agents usually incorporate three instruments for electricity trading:the pool (spot
market operated by the system operator), bilateral contracts and the derivative securities. In
the pool, agents submit bids, consisting of a set of quantities at certain prices for the following
day. And the system operator clears the market and announces the set of clearing prices for
the next day. These daily bids should include the price forecasts as well as production and
consumption plans of the agents. Successful forecasts of the following day’s price can help
producers develop revenue maximizing strategies, or maximization of consumers’ utility as it
is also stated in Contreras et al. (2003) [30].
8
Moreover, agents need medium term quantity and price projections in orderto determine
bilateral contract terms. With the help of reliable daily price forecasts, producers or consumers
can price these bilateral contracts more efficiently. Also for market players who want to be
hedged against the price volatility, expected future spot prices are essential in valuation of
derivative instruments.
We will model the daily spot prices (equally weighted average of the hourly equilibrium
prices of each day), instead of the hourly system balancing prices. Mainstay of our approach
is the observation done by Meyer-Brandis et al. (2007) in [57], which states that hourly/ half
hourly quoted electricity prices cannot be seen as a time series, since in mostof the electricity
markets, the delivery prices for all 24 hours of a given day are pre-determined by the system
operator simultaneously on the previous day (day-ahead prices). Afterthis determination,
day-ahead market closes and a limited number of transactions take place randomly in real
time balancing market. Thus, there is no causality relationship between different hourly prices
on the same day, and hourly electricity data should rather be seen as 24 dependent daily series
than a single series of hourly prices. With this line of reasoning, Meyer-Brandis et al. (2007)
[57] suggest the following model for hourly electricity prices:
Xht = Xh
t f (t,h) + ǫht (2.1)
where
• Xht , h ∈ 1, . . . ,24 is the electricity price on dayt and hourh
• Xht is the common factor for dayt
• f (t,h) is a slowly varying intradaily pattern depending both on the dayt and the hourh
• ǫht is a white noise process
In this setting, most of the variability observed in electricity price, as well as allinteresting
statistical features, e.g. mean reversion, are assumed to be contained in theaverage daily price
series.
9
2.3 Spot Electricity Models in Literature
In general, electricity price models can be grouped under three generaltopics. The first ap-
proach is based on econometric time series models. Apart from the basic autoregressive (AR)
and autoregressive moving average (ARMA) specifications, a wide range of alternative mod-
els have been proposed. This long list of models includes: autoregressive integrated moving
average (ARIMA) and seasonal ARIMA models (Zhou et al. (2004) [80]), autoregressions
with heteroscedasticity (Garcia et al. (2005) [41]) or with heavy tailed (Weron (2005) [77])
innovations, AR models with exogenous variables (ARX models), AR and ARXmodels with
thresholds (Misiorek et al. (2006) [58]), regime switching regressionswith fundamental vari-
ables (Weron (2006) [78]) and mean reverting jump diffusions (Knittel et al. (2005)[50]).
The second group of models consists of fundamental/ forward based models, the futures
prices and their relation with the spot electricity prices are the main focus of thestudy, and the
dynamics of the whole futures price curve is modeled by using Heath-Jarrow-Morton (HJM)
framework. See for instance, [3], [17], [27], [51]. A general discussion of HJM-type models
in the context of power future is given in Benth et al. (2008) [10]. Theydedicate a substantial
part of their analysis to the relation of spot, forward and swap price dynamics. However, since
non-storability of the electricity causes the break up of no arbitrage condition between spot
and future markets, futures prices do not reveal any information aboutprice dynamics on a
daily timescale, but they can provide only a poor approximation to the complex structure of
spot market prices.
The classical starting point for the commodity price modeling is the Schwartz one-factor
model [70] , which is an extension of geometric Brownian motion allowing for mean rever-
sion:
St = S0 exp(Xt), (2.2)
dX(t) = α(µ − X(t))dt+ σdB(t), (2.3)
whereBt is a standard Brownian motion,σ is the volatility of the process andα is the speed of
the process reverting to its long term meanµ. Many electricity price models use this process
or its variants as building blocks. For instance, spot price models proposed in Lucia et al.
10
(2002) [53] are given (2.4) and (2.5) by
St = h(t) + Xt, (2.4)
St = exp(h(t) + Xt), (2.5)
whereSt is the spot price,Xt is an Ornstein-Uhlenbeck process andh(t) is a deterministic
component, which is used to account for the seasonal effects. Weekly seasonality due to
higher weekday prices than weekend prices, and spot prices’ cyclical behaviour throughout
the year are all included in spot price dynamics by the termh(t). Benth et al. (2008) [10]
defined (2.4) type models as arithmetic models and models with the form given in (2.5) as
geometric models in which logarithmic prices can be characterized by an Ornstein-Uhlenbeck
process.
Spot price models can be examined under two subgroups: single factor and multi-factor mod-
els. In a single factor model the spot price itself is a Markov process while ina multi-factor
model the spot priceSt = g(Xt1, ...,X
tk) is a function of a multi-dimensional Markov process.
Hereg : Rk → R+ and sinceg is not one-to-one, these models have both unknown and hid-
den components. Not only Lucia and Schwartz (2002) [53], but also Cartea et al. (2005) [26],
Barlow (2002) [4], Geman et al.(2005) [42] proposed single factor models. Many of these
models, unlike Lucia and Schwartz’s, take price spikes into account too. With an additional
jump term included in an Ornstein-Uhlenbeck process, Cartea et al. (2005) [26], improved
their models to cover also the price spikes:
logSt = h(t) + Yt, (2.6)
dY(t) = −αYtdt+ σdBt + JtdNt, (2.7)
whereBt is a Brownian motion,h(t) is assumed to capture the seasonal patterns of the spot
price and the termJtdNt enables the process to have discrete random spikes which are com-
bination of a Poisson processNt and a jump size distributionJt. In (2.7) the processdNt
is approximated by a Bernoulli process with parameterldt and Jt is assumed to be lognor-
mally distributed. Cartea et al. (2005) [26] apply this one factor jump diffusion model spot
electricity markets in England and Wales. Unfortunately, unless the price dataseries is not
long enough, few price spike observations lead to difficulties in estimation of the parameters.
Moreover, one factor models including a jump process are expected to have a high speed of
mean reversion. Otherwise, jumps can have permanent effects on the price levels.
11
Barlow (2002) [4] introduced a nonlinear Ornstein-Uhlenbeck model forspot power prices.
In this model, the price is obtained by equalization of the demand level with a deterministic
non-linear supply function to take price spikes into account. He used the inverse function of
the Box-Cox transformation:
St =
fα(Xt), if 1 + αXt > ǫ0,
ǫ1/α0 , if + αXt ≤ ǫ0,
dXt = −λ(Xt − α)dt+ σdB(t), (2.8)
where fα(x) = (1+ αx)1/α, α , 0, ǫ1/α0 is assumed to be the maximum price level taken as a
constant andf0(x) = ex.
Whenα = 0, an exponential Ornstein-Uhlenbeck process is retrieved forSt. The caseα = 1
yields a regular Ornstein-Uhlenbeck process. The model is fitted to Albertaand California
markets using the maximum likelihood estimation.
Geman et al. (2006) [43] also proposed a mean reverting jump process, where the long term
mean level is assumed to represents the marginal cost of electricity production, which can be
a constant, a periodic function or a periodic function with a trend. Random moves around the
average trend represent the temporary supply demand imbalances in the network. The model
assumes that the natural logarithm of electricity price is described by a stochastic differential
equation of the form
dE(t) =[h(t) + Θ(µ(t) − E(t−))
]dt+ σdW(t) + f (E(t−))dJ(t); (2.9)
whereh(t) is a deterministic seasonality function,Θ is a positive parameter representing the
average variation of the price per unit of shift away from the trend, meanreversion level. The
process reverts back to a deterministic mean level rather than the stochastic pre-spike value.σ
is the volatility attached to the Brownian shocks. The last term in the equation represents the
discontinuous part of the model featuring price spikes. This effect is characterized by three
quantities: occurrence, direction and size of jumps. The functionf assumes±1 depending on
the level of the spot prices:
f (E(t−)) =
+1, ifE(t) < τ(t),
−1, ifE(t) ≥ τ(t).
Steps followed by the authors in the estimation of model (2.9) are summarized below.
12
1. The first step is the detection of jumps in the raw market data. Authors observed that
log returns tend to cluster close to either their average mean or to the largest observed
values. In other words, data suggest that either there is a jump, in which case variation
due to the continuous part is negligible; or there is no jump and variations in the price
level are due to the continuous part of the process. It is concluded thatby using a
price change thresholdT, main driver of the changes observed in price levels can be
identified.
2. An affine function and two cosine functions with 12 and 6 month periods are used to
estimate the deterministic dynamics of the jump free price series:
µ(t;α, β, γ, ǫ, δ, ζ) = α + βt + γ cos(ǫ + 2πt) + δ cos(ζ + 4πt). (2.10)
The first term represents the fixed cost linked to the production of powerwhile second
one drives the long run linear trend in the total production cost. The overall effect of
the third and the fourth terms is a periodic path displaying two maxima per year.
3. The third step is the determination of the jump intensity functionλ: Let
λ(t) =
(2
1+ |sin[π(t − τ)/k]|− 1
)d
. (2.11)
With this model, jump occurrence exhibits peaking levels at multiples ofk years begin-
ning at timeτ. The powerd allows us to adjust the dispersion of jumps around peaking
times and it is included among parameters to be estimated.
4. The probability distribution of the jump sizes is assumed to be a truncated version of
the exponential distribution with parameterθ:
p(x; θ, ϕ) =θ exp(−θx)
1− exp(−θϕ). (2.12)
The model parameterθ andϕ is estimated by using the log likelihood function.
5. The estimation of the constant Brownian volatility over observation dates is as follows;
σ =
√√√n−1∑
i=0
(∆E(ti)
)2, (2.13)
13
where(∆E(ti)
)2represents the square of the continuous parts of observed logarithmic
price variations between consecutive days.
The last two models neither include the convenience yield as a factor, nor considers the val-
uation of the futures contracts (or any other kind of derivative). The single factor models are
quite tractable and their parameters are relatively easy to estimate. However,they have a seri-
ous constraints: they cannot explain the relation between the spot and the futures prices well
enough. Spot price models depending on more than one factor help to develop the relation
between spot and future commodity prices.
For instance, Brennan et al. (1985) [19] assumed that the spot copper price follows a geomet-
ric Brownian motion and incorporated a convenience yield to their model that isproportional
to the spot price:
dSt = µStdt+ σStdz, (2.14)
C(S, t) = cS. (2.15)
The idea of a constant convenience yield holds only under restrictive assumptions, since the
theory of storage is rooted in an inverse relationship between the convenience yield and level
of inventories. Gibson et al. (1990) [44] took an important step to a more realistic model
of economy by introducing a stochastic convenience yield rate. The spot price St of the
commodity is described by a geometrical Brownian motion and the convenience yield rateδt
is described by an Ornstein-Uhlenbeck process with equilibrium levelα and the rate of mean
reversionκ:
dSt = (µ − δt)Stdt+ σ1Stdz1, (2.16)
dδt = κ(α − δt)dt+ σ2dz2, (2.17)
dz1dz2 = ρdt. (2.18)
Significant contributions to this kind of models have been made by Schwartz (1997) [70].
He reviewed one and two factor models and developed a three factor modelunder stochastic
convenience yield and interest rates. Inclusion of the interest rate as a third factor makes
14
forward and futures prices different:
dSt = (rt − δt)Stdt+ σ1Stdz1, (2.19)
dδt = κ(α − δt)dt+ σ2dz2, (2.20)
drt = a(m− rt)dt+ σ3dz3, (2.21)
dz1dz2 =ρ1dt, dz2dz3 = ρ2dt, dz1dz3 = ρ3dt. (2.22)
This model was originally developed for copper and oil market. Kalman filter algorithm was
used to estimate the parameters in the models.
In [53], Lucia and Schwartz (2002) analyzed the Nordic power marketand model the spot
price as
St = h(t) + Xt + Yt, (2.23)
dXt = −λXtdt+ σXdWX, (2.24)
dYt = −µtdt+ σYdWY, (2.25)
dWXdWY = ρdt. (2.26)
The functionh(t) is deterministic, and it is intended to capture the predictable component
in the spot price, i.e, seasonal effects. This function distinguishes between weekdays and
includes a monthly seasonal component employing dummy variables. The idea of this model
is to have a non-stationary process for the long term equilibrium price levelY and short term
mean reverting componentX. They estimated all the parameters simultaneously by nonlinear
least squares method.
The multi-factor models described so far do not capture one of the most characteristic features
of the electricity prices, jumps or spikes. Several authors, such as Deng(2000) [33] and
Villaplana (2004) [76] extend these models with both diffusion and jumps. In the work of
Villaplana, power prices are modeled according to non-observable state variables that account
for the short term movements and long term trends in electricity prices:
ln St = h(t) + Xt + Yt, (2.27)
dXt = −κXXtdt+ σXdW1 + JudN(λu) + JddN(λd), (2.28)
dYt = −κY(µ − Yt)dt+ σYdW2, (2.29)
dW1dW2 = ρdt. (2.30)
15
The jump components are characterized byN(λu) andN(λd):, Poisson processes with intensi-
tiesλu andλd respectively and by random jumps of sizesJu andJd with a specified distribution
like Gaussian or exponential.
Deng (2000) [33] and Villaplana (2004) [76] set their models in affine jump diffusion frame-
work which enabled them to use transformed results of Duffie et al. (2000) [37] to derive
tractable closed form solutions for a variety of contracts. Deng proposed more sophisticated
mean reverting jump diffusion models with deterministic/ stochastic volatility and regime
switching. This seems a good way of dealing with the dramatic changes in the electricity
prices. However, both of the authors concluded that the trajectories produced by their models
are fairly different from the ones observed in the market.
Cartea et al. (2005) [26] built a model for wholesale power prices defined by two state vari-
ables (demand and capacity) and calculate the forward premium:Dt and Ct representing
demand and capacity:
Dt = fD(t) + XDt , (2.31)
Ct = fC(t) + XCt , (2.32)
where fD, fC are deterministic functions andXDt , XC
t are independent Ornstein-Uhlenbeck
processes. They constructed the spot price process as
St = β exp(α)Dt + γCt. (2.33)
Models discussed in Benth et al. (2008) [11] constitute the starting point ofthis thesis. A
brief summary of their theoretical framework is given below.
• The spot markets of electricity quote prices on an hourly or half hourly basis. Thus, it
will not make sense to talk about spot price of electricity at any time t. On the other
hand, if there exists an electricity futures market, electricity contracts (settledaccording
to the hourly prices) are traded in a continuous market in the sense that the actors can
buy or sell at any time as long as they find a counterpart in the market. Hence, contrary
to most other commodity markets where there is a liquid trading in both spot and future
/ forwards, we face the situation of a discrete spot and a continuous time futures market.
16
• A continuous time stochastic processS(t), represents the unobserved instantaneous spot
price of electricity at timet with delivery time [t, t + dt). The processS(t) can be
regarded as the price market participants would pay if they could buy electricity at time
t with infinitesimal delivery time. In the market, we observe the price of electricity with
delivery over a specified hour. Following some simple calculations, authorsreached the
conclusion that:
Sdi = S(tdi ), (2.34)
whereSdi is the price of electricity for theith hour of dayd andS(tdi ) is the instantaneous
price at the beginning of theith hour of dayd. Since we have the spot price at time
momentstdi , we have the observations of an underlying continuous time spot price
process of electricity.
• Representing the logarithmic prices or the prices itself by a series of Ornstein-Uhlenbeck
processes allows us to model different speeds of mean reversion and to incorporate a
mixture of jump and diffusional behavior of the prices. Price spikes can be modeled by
an Ornstein-Uhlenbeck process having a low frequency of big jumps with fast mean re-
version, while more normal price variations are represented by a slower mean-reverting
process driven by a Brownian motion.
• Seasonality in jumps is captured by using an independent increment process in the jump
model.
Geometric models are formalized as:
ln S(t) = ln∆(t) +m∑
i=1
Xi(t) +n∑
j=1
Yi(t), (2.35)
dXi(t) = (µi(t) − αi(t)Xi(t))dt+p∑
k=1
σik(t)dBk(t), (2.36)
dYj(t) = (δ j(t) − β j(t)Yj(t))dt+ ν j(t)dI j(t), (2.37)
where the deterministic seasonal price level is modeled by the function∆(t), which is as-
sumed to be continuously differentiable. An additional drift term can be imposed by the jump
components, since they are not assumed to be martingales. For instance, theoccurrence of
price spikes should add an amount to the overall expected spot price in excess of the seasonal
function.
17
Arithmetic models are given as
S(t) = ∆(t) +m∑
i=1
Xi(t) +n∑
j=1
Yi(t). (2.38)
Independent increment processes are also assumed to be independent. The arithmetic models
can lead to negative prices, although this kind of price movements can be observed: they are
rare events. A class of arithmetic models having zero probability of negativeprices can be
constructed by supposingm= 0 and the seasonality function∆(t) is a floor towards which the
processesYj revert. Moreover, it is assumed that the probability of negative jumps is zero.
Then, under the assumption that∆(t) is positive, the spot price model produces only positive
prices. Hence, there are negative and positive price fluctuations arising from a combination
of downward mean reversion and upward jumps. If the sum of the jumps over the increment
is stronger than total contribution by mean reversion, we observe a random price increase. A
price decay is observed otherwise.
Authors provide closed form solutions for forwards and options on forwards. This model cou-
pled with a good description of seasonality provides a precise characterization of electricity
spot price behavior. Although the model seems to capture the stylized facts of the spot price
market such as mean reversion, seasonality and price spikes, there areno precise statistical
analysis about the quality of the model. However, they suggested the particlefilter as a possi-
ble solution for estimation of the parameters in the model. As a result, parameter estimation
for the model appears to be a significant challenge.
The need for a better fit of model forecasts with observations leads to morecomplicated
models with hidden variables and multiple regimes. However, with an increasing number of
parameters, the estimation process also gets complicated. Filtering techniques are one of the
widely used methods in parameter estimation of the jump processes. Pirino et al. (2010) [65]
used an iterative threshold filtering in identification of spikes in their univariate jump model
and use the seperated processes for parameter estimation. Their model is nonparametric in the
sense that it is free from parametric model assumptions and flexible in capturing the dynamics
of the data. The estimation is performed in two steps. In the first step, spikes are identified by
means of an iterative filtering technique. Then, series of spikes are usedto estimate a seasonal
jump intensity function.
18
This work proposes a combination of Benth et al. (2008) [11] multi-factor models and Pirino
et al. (2010) [65] threshold spike detection idea. In our model, three jump processes with-
out predetermined jump size distributions and a Brownian motion are combined witha de-
terministic seasonality term where two of the jump processes are mean reverting. While the
Brownian motion captures the daily regular price movements, pure jump process models price
shocks which have long term effect, and further the two Ornstein-Uhlenbeck type processes
with different mean reversion speeds capture price shocks that have short term effects. One
of the mean reverting processes is assumed to model price shocks that revert back in the next
observation (price spikes) and the other is assumed to model price shocksthat take a few days
to fade away (semi-spikes). An iterative threshold derived by using estimated volatility with
GARCH(1,1) is used to filter the price jumps. Although in Mancini (2009) [54], a threshold in
order to separate jumps form Brownian motion is proposed and in Mancini etal. (2010) [55]
GARCH(1,1) volatility estimation is suggested to be used in calculation of threshold, none of
the models investigated in these researches include as many factors as our model have. On
the other hand, instead of Pirino et al. (2010) [65] which use a kernel based estimator for
threshold, we use GARCH(1,1).
19
CHAPTER 3
THE SPOT PRICE MODEL
Given the filtered probability space (Ω, F, Ftt∈[0,T] ,P) where the filtrationFtt∈[0,T] satisfies
the usual conditions i.e., the filtrationFt is right continuous andF0 contains allP-null sets.
Moreover,Wt is a standard Brownian motion andJ(1)t , J(2)
t and J(3)t are finite activity pure
jump processes. A process is said to have finite activity if almost all paths of the process
have only a finite number of jumps along finite time intervals. For the formal definition of
the finite activity jump processes, see Appendix A.1. The simplest examples include the
are Poisson and compound Poisson processes. Levy measureκ(i) for i = 1,2,3 satisfies
κ(i)(dt,dx) = λ(i)dtF(i)(dx), whereλ(i) represents the expected number of jumps in the unit
time interval,F(dx) represents the jump size distribution. Following Benth et al. (2008)
[11], we assume that there exits a continuous electricity price process,St governed by the
exponential price equation given in (3.1).
St = eΛt+Pt , (3.1)
whereΛt is the deterministic seasonality function andPt, governs deseasonalized logarithmic
prices process. Although observing negative prices is possible in spotelectricity markets, we
rule out this possibility by using an exponential price model, since it is a rare event. There
will be three jump factors and a diffusion term in the model. The dynamics ofPt is:
dPt = σtdWt + dJ(1)t + dYt + dZt, (3.2)
dYt = −νYtdt+ dJ(2)t , (3.3)
dZt = −βZtdt+ dJ(3)t , (3.4)
whereJ(1)t , J(2)
t andJ(3)t are compound Poisson processes with the corresponding jump inten-
sitiesλ(1)t , λ(2)
t andλ(3)t , N(1)
t , N(2)t andN(3)
t are the respective jump counting Poisson processes.
20
We do not make any initial assumptions about the distribution of the jump processes. How-
ever, it is assumed thatWt, J(1)t , J(2)
t and J(3)t are mutually independent withJ(i)
0 = 0 for
i = 1,2,3 and stochastic volatilityσt is a progressively measurable processes.
Therefore, stochastic processPt is composed of a stochastic volatility Brownian motion, as-
sumed to capture regular daily movements of spot prices. A pure jump process J(1)t repre-
senting structural changes that have long term effects on electricity prices like privatization,
technological advances etc. the two non-Gaussian Ornstein-Uhlenbeckprocesses with jump
processesJ(2)t andJ(3)
t have different dissipation rates for price shocks created by the jumps.
In this settingYt, spike process, represents any abrupt change in price level which isreversed
quickly, like a price shock affecting the price levels only a couple of hours rather than the
whole day. On the other hand,Zt, semi-spike process, having a lower mean reversion period,
assumed to capture factors affecting price level in spot market longer than spike process, like
extreme weather conditions which can be effective for a few days; or supply chain problems,
system failures that take a few days to resolve.
Solving (3.2) yields
Pt =
∫ t
0σsdWs+
∫ t
0dJ(1)
s + Y0e−νt +∫ t
0e−ν(t−s)dJ(2)
s + Z0e−βt
+
∫ t
0e−β(t−s)dJ(3)
s , (3.5)
where we assumeP0 = Y0 + Z0. If we express jump components by summation:
Pt =
∫ t
0σsdWs+
N(1)t∑
k=1
∆kJ(1) + Y0e−νt +N(2)
t∑
k=1
e−ν(t−τ(2)k )∆kJ(2) + Z0e−βt
+
N(3)t∑
k=1
e−β(t−τ(3)k )∆kJ(3), (3.6)
whereτ(i)k is the random time of thekth jump of theith jump process. Here∆kJ(i) := J(i)(τ(i)k )−
J(i)(τ−k(i)) becomes the amount of thekth jump of theith jump process. On the other hand,
∆J(i)tn := J(i)
tn − J(i)tn−1
. The latter difference formula holds for all of the processes in (3.2), (3.3)
and (3.4). Moreover, for simplification, it is assumed that there can only beone jump in each
observation interval.
21
Returning back to the process lnSt
ln St = Λt +
∫ t
0σsdWs+
N(1)t∑
k=1
∆kJ(1) + Y0e−νt +N(2)
t∑
k=1
e−ν(t−τ(2)k )∆kJ(2) + Z0e−βt
+
N(3)t∑
k=1
e−β(t−τ(3)k )∆kJ(3). (3.7)
It is well known that both Brownian motion and the Levy processes are semimartingales.
Moreover according to Protter (2004) [66] (Chapter 5, Theorem 19)the stochastic integral
process∫ t
0HsdXs is a semi-martingale ifH is a left continuous right limited (caglad) process
and X is a semi-martingale. Then assuming thatσt is a caglad process each of the terms
in Pt are semi-martingales. Again referring to Protter (2004) [66](Chapter 2,Theorem 1);
the set of semi-martingales is a vector space for the given probability space. ThereforePt is
also a semi-martingale. In the following sections we discuss the modules of the estimation
procedure and our spot price model.
3.1 Deterministic Part
It is assumed that logarithmic spot prices are sum of two independent components, a pre-
dictable deterministic componentΛt and a stochastic componentPt. As already discussed,
electricity prices, heavily affected by weather conditions, exhibit a cyclical behavior through
the year. Not only the demand side but also the supply side may show seasonal variations,
like the hydro units heavily dependent on precipitation and snow melting. Besides the annual
seasonality, like it is shown in Figure 2.2 there is a significant day of the weekeffect in the
spot prices, due to varying electricity demand.
Typically we do not observeSt continuous in timet, but in the form of discrete observations.
Working with discrete observation series, total observation period[0,T] is divided intoN
equal intervals with lengthh = T/N. Thus,Stn = Snh for n = 1, ...,N.
Although our intention at the beginning was modelingΛt, the deterministic function as a
summation of a constant, a linear trend (aiming to capture the inflationary pressures on the
electricity prices), and sinusoidal weekly and annual functions, but it isobserved that weekly
sinusoidal cycles are insufficient in the elimination of weekly autocorrelation pattern, there-
fore we divide the procedure into two: inspired by Weron (2006) [78] and Mayer et al. (2011)
22
[56], with the procedure summarized below, persistent weekly autocorrelation in the logarith-
mic price series is eliminated.
Procedure for elimination of persistent weekly autocorrelation
1. Data is smoothed using a moving average filter:
m(tn) =17
(ln Stn−3 + . . . + ln Stn+3).
2. Each days deviation from the corresponding moving average is calculated. Then for
each day of the week average of that days deviation is computed. For instance, if the
first day of your observation series is Monday the deviation is calculated as follows:
w(monday) =1k
k−1∑
j=0
[ln(S(7 j+1)h) −m((7 j + 1)h)],
which is the corresponding weekly seasonality term, wherek is the number of Mondays
in the observation set.
3. Summation of daily seasonalities is normalized to ensure that they add up to zero for
each week.
After subtracting the estimated daily effects from logarithmic price series, coefficients of an-
nual seasonality function with linear trendα0+α1tn+α2 cos((2π(tn−α3))/365) are estimated,
wheren = 1, ...,N. Therefore, the deterministic seasonality function becomes:
Λtn = w(tn) + α0 + α1tn + α2 cos
(2π (tn − α3)
365
). (3.8)
3.2 Jump Detection with Threshold Method
From now on, we are assuming that deterministic seasonal part is removed and we are working
with deseasonalized logarithmic price processPt. In the related literature it is not possible to
find a common definition for a price spike. In a broader sense, it is widely accepted that price
jumps are movements in the price level that surpass a threshold and price spikes are composed
of price observations that exceeded threshold level for a short period of time. The key variable
that should be found is the level of this threshold. The existing literature provides some
23
answers to this question like detecting jumps using boundaries implied by normal distribution
(Borovkova et al., 2004 [16]), filtering raw data using different deterministic thresholds and
selecting the threshold level which gives the best calibrated model in terms ofapproaching
to the kurtosis of the original daily price variations (Geman, 2006 [43]), orusing wavelet
techniques (Stevenson, 2001 [71]). Our filtering technique which is based mainly on the
work of Mancini (2009) [54] is summarized below.
In a continuous time setting, considering the asymptotic theory, for sufficiently small intervals,
jumps can be detected as the increments of the process which are numerically too large to
come from a continuous variation. In order to test this statement, it is necessary to determine
how much a continuous process can move on a specified time interval or to determine the
distribution of the largest increments generated by continuous component of the process. The
stochastic continuous component in our price process is the stochastic volatility Brownian
motion. The first step is to find an upper limit on these continuous movements, in other words
how much a Brownian motion can move on a given time intervalh. Paul Levy’s law for the
modulus of continuity of Brownian motion paths implies that
limh→0
supn∈(1,...,N)
|∆Wtn |√2h log 1
h
≤ 1 almost surely(a.s.)
When we adopt Theorem 1 of Mancini (2009) [54] to our stochastic processPt, we get the
following result.
Theorem 3.2.1 Identification of the intervals where no jumps occurred:Consider the sys-
tem given in (3.2), (3.3) and (3.4) where all the jump processes, J(1), J(2) and J(3) are mutually
independent, finite activity jump processes such that f or all tn ∈ [0,T], P(∑3
i=1∆N(i)tn , 0,
∑3i=1∆J(i)
tn = 0 = 0. Suppose also that
1. lim suph→0
supn∈(1,...,N) |∫ tntn−1σ2
sds|h ≤ M(ω) < ∞ a.s.,
2. The threshold level r(h) is a deterministic function of lag h between the observations,
such thatlimh→0 r(h) = 0 and limh→0h log 1
hr(h) = 0.
Then for P− almost allω, there ish(ω) > 0 such that f or every h≤ h(ω) n = 1, ...,N:
I(∆Ptn)2≤r(h)(ω) = I∑3i=1∆N(i)
tn =0(ω). (3.9)
24
Proof: Like Mancini (2009) [54], the proof of the theorem is divided in two steps. Firstly, we
show that ,I∑3i=1∆N(i)
tn =0(ω) ≤ I(∆Ptn)2≤r(h)(ω) a.s. for smallh. In the second step we prove
that I∑3i=1∆N(i)
tn =0(ω) ≥ I(∆Ptn)2≤r(h)(ω) a.s. for smallh.
Before starting; we prove that stochastic integral∫ t
0σsdWs is a bounded process. Assuming
thatσt is a continuous process;∫ t
0σsdWs is a time changed Brownian motion. Dambis (1965)
[32] and Dubins and Schwartz (1965) [36] showed that ”any continuous martingale is a time
changed Brownian motion”. In 1978, Monroe [59] extended this result toa more general set-
ting that ”any semi-martingale is a time changed Brownian motion”. More preciselythere is a
Brownian motionW with respect to the filtrationFtt≥0 such that for eacht ≥ 0,ω(to)[X]t(ω)
is stopping time andXt =W[X]t , whereXt is a local martingale. In particular, ifW is a Brow-
nian motion, andσ is square integrable then we letXt =∫ t
0σsdWs. Since [X]t =
∫ t
0σ2
sdswe
have∫ t
0σsdWs =
d W∫ t0 σ
2sds. Considering
supn∈(1,...,N)
∣∣∣∣∫ tntn−1σsdWs
∣∣∣∣√
2h log 1h
≤ supn∈(1,...,N)
∣∣∣∣∣W∫ t0 σ
2sds−W∫ tn−1
0 σ2sds
∣∣∣∣∣√
2∫ t
tn−1σ2
sdslog 1∫ ttn−1σ2
sds
supn∈(1,...,N)
√2Mh log 1
Mh√
2h log 1h
supn∈(1,...,N)
√2∫ t
tn−1σ2
sdslog 1∫ tntn−1σ2
sds
√2Mh log 1
Mh
. (3.10)
When we take the limit of the above process ash → 0 the first argument is smaller than or
equal to 1 due to modulus of continuity as the time index is the stochastic integral instead of
h. Due to assumption 1 and monotonicity of the functionx log(1/x) in the neighborhood of 0
second argument is also bounded and therefore (3.10) is bounded ash→ 0
Now we may carry out the steps to prove the theorem:
1. For eachω, J0,h =n ∈ 1, ...,N :
∑3i=1∆N(i)
tn = 0. To show that for smallh,
I∑3i=1∆N(i)
tn =0(ω) ≤ I(∆Ptn)2≤r(h)(ω), it is sufficient to prove that for smallh,
supJ0,h(∆Ptn)
2 ≤ r(h) holdsa.s.. On the setJ0,h for i = 1,2,3, we haveN(i)tn = N(i)
tn−1,
25
∆Ptn =
∫ tn
tn−1
σsdWs
︸ ︷︷ ︸An
+Y0e−νt(1− eνh) +
N(2)tn−1∑
k=1
e−ν(tn−τ(2)k )(1− eνh)∆kJ(2)
︸ ︷︷ ︸Bn
+ Z0e−βt(1− eβh) +
N(3)tn−1∑
k=1
e−β(tn−τ(3)k )(1− eβh)∆kJ(3)
︸ ︷︷ ︸Cn
. (3.11)
The square of∆Ptn is summation of six arguments. Now we check each of these argu-
ments relative to the threshold functionr(h) ash→ 0:
suptn∈J0,h
(∆Ptn)2 ≤ sup
tn∈J0,h
|An|2 + suptn∈J0,h
|Bn|2 + suptn∈J0,h
|Cn|2 + suptn∈J0,h
(2|An||Bn|).
+ suptn∈J0,h
(2|An||Cn|) + suptn∈J0,h
(2|Bn||Cn|)
Considering|Bn||Cn| we obtain that
limh→0
supJ0,h
|Y0e−νt(1− eνh
)+
∑N(2)(tn−1)k=1 e−ν(tn−τ
(2)k )(1− eνh)∆kJ(2)|
√h log 1
h
|Z0e−βt(1− eβh) +∑N(3)(tn−1)
k=1 e−β(tn−τ(3)k )(1− eβh)∆kJ(3)|
√h log 1
h
h log 1h
r(h)
is equal to 0.|Bn|2 and|Cn|2 follow the same line of reasoning and goes to 0 ash→ 0.
On the other hand we have already showed that suptn∈J0,h|An|/
√2h log 1
h ash → 0 is
bounded. Therefore suptn∈J0,h|A2
n|/√
2h log 1h is also bounded. The|An||Bn| yields;
limh→0
supJ0,h
|∫ tntn−1σsdWs|
√2h log 1
h
×|Y0e−νt(1− eνh) +
∑N(2)(tn−1)k=1 e−ν(tn−τ
(2)k )(1− eνh)∆kJ(2)|
√2h log 1
h
×2h log 1
h
r(h).
(3.12)
We know that the first factor is bounded and the third factor goes to 0. Second factor
also goes to 0 by L’Hospital rule. Therefore, cross terms|An||Bn| and|An||Cn| relative to
the threshold, go to zero in the limit. As a result, we conclude that ifn ∈ J0,h ash→ 0
supJ0,h(∆Ptn)
2 ≤ r(h).
26
2. In order to establish the second inequality, it is assumed that for anyω, J1,h = n ∈ (1, ...,N) :∑3
i=1∆N(i)tn , 0. It is sufficient to show that for smallh, inf J1,h(∆Ptn)
2 ≥ r(h) a.s. to prove
that f or every n, I∑3i=1∆N(i)
tn =0(ω) ≥ I(∆Ptn)2≤r(h)(ω) holds almost surely.
Working in the subsetJ1,h in addition to the terms in the previous step there are also
jump terms;∆N(1)(tn)J(1)+e−ν(tn−τN(2)(tn))∆N(2)(tn)J
(2)+e−β(tn−τN(3)(tn)) ∆N(3)(tn)J(3). Consider
that τN(tn) ∈ (tn−1, tn] as h → 0, τN(tn) → tn andτN(t−n ) → tn−1 therefore discounting
terms for the mean reverting jump processes collapse to 1 ash → 0. Moreover, with
the assumption that there can only be one jump on each time interval, we get
limh→0
infJ1,h
(|∆N(1)(tn)J(1) + e−ν(tn−τN(2)(tn))∆N(2)(tn)J
(2)
r(h)
+e−β(tn−τN(3)(tn))∆N(3)(tn)J(3)|)2
r(h)= ∞, (3.13)
Since the jump processes we consider are finite processes, the numeratoris a fixed
amount while the denominator goes to 0 ash→ 0. Let us look to the cross terms
limh→0
infJ1,h
(|∆J(1)(τN(1)(tn)) + e−ν(tn−τN(2)(tn))∆N(2)(tn)J(2)
r(h)
+e−β(tn−τN(3)(tn))∆N(3)(tn)J(3)|)
r(h)
[|An|√r(h)+|Bn|√r(h)+|Cn|√r(h)
]. (3.14)
The summation in the brackets is bounded at the limit and the first term goes to 1 as
h→ 0. As a result,
limh→0
infJ1,h
(∆Ptn)2
r(h)= ∞. (3.15)
this completes the proof.
3.3 Determining The Threshold
In Mancini et al. (2010) [55]hα, α ∈ (0,1), is used as the deterministic, constant threshold
function. In their model for the interest rates Mancini and Reno stated thatTheorem 3.2.1
27
holds even if the threshold functionr(h) varies with time according to
rt(h) = ct r(h), (3.16)
where ˜r(h) satisfies the threshold conditions andct is an a.s. bounded stochastic process which
is also bounded away from 0. Following Mancini and Reno; Pirino et al. (2010) [65] adapted
the following threshold in their discrete time electricity model, iterated on the integerZ;
θZti = c2v
∑Lj=−L, j,−1,0,1 K( i
L )∆i+ jP2I∆i+ j P2≤θZ−1
ti+ j
∑Lj=−L, j,−1,0,1 K( i
L )∆i+ j I∆i+ j P2≤θZ−1
ti+ j
(3.17)
where the parametercv sets the number of the standard deviations after which an observation
is considered to be above the threshold. The value of thecv potentially determines the number
of detected jumps, however it is denoted that since in the case of electricity price, jumps are
too large hence the choice of this value is mostly uninfluential.
Corsi et al. (2009) [31] state that, in applications, it is natural to scale threshold function with
respect to local spot return variancert = c2σt, where ˆσt is an auxiliary estimator ofσt andc
is a positive constant.
We use the multiple of estimated stochastic volatility as a threshold function. Althoughthe
proposed model for the electricity prices is in continuous time, since our priceobservations
are discrete, we will use a discrete GARCH(1,1) process for the stochastic volatility model-
ing. This is not an arbitrary choice: in Nelson (1992) [62], it is stated thatARCH type models
are remarkably robust to certain types of misspecification such that as longas the process
is well approximated by a diffusion, broad classes of ARCH models provide consistent esti-
mates of the conditional variance. Nelson (1992) [62] especially stresses that in this context
the term’estimate’corresponds to its use in the filtering literature rather than the statistics
literature, i.e., the GARCH(1,1) model with fixed parameters produces estimatesof the true
variance vector at each time point in the same sense that a Kalman filter produces estimate of
unobserved state variables in a linear system. Let us assume that there areno jumps at time
t, then our instantaneous deseasonalized logarithmic price process is sum of a deterministic
mean reversion part of the previously observed jumps and initial values ofprocessYt andZt
described byµt plus a diffusion component by standard Brownian motionBt:
28
dPt = dµt + σtdBt. (3.18)
Using a sequence of discrete time observations, with a fixed observation interval h, we ap-
proach to continuous time stochastic volatility process with a GARCH process. Let (ǫnh)n∈N0
be an i.i.d. sequence of standard normal random variables, letξh, δh ≥ 0 andηh > 0 and let
(P0h, σ20h) be starting random variables independent of (ǫnh)n∈N0. Then (Pnh−P(n−1)h, σ
2nh)n∈N0
defined recursively by
Pnh = P(n−1)h + ∆nµh + h1/2σnhǫnh, (n ∈ N) , (3.19)
σ2nh = ξh + (ηhǫ
2(n−1)h + δh)σ2
(n−1)h, (n ∈ N) . (3.20)
Equations above define a GARCH(1,1) process. Here∆nµh = Bn + Cn, whereBn andCn
are given in (3.11). Note that, (Pnh, σ2nh)n∈N0 is embedded into a continuous time process
(Pt,h, σ2t,h)t≥0 by defining
Pt,h := Pnh, σ2t,h := σ2
nh, nh≤ t < (n+ 1)h. (3.21)
Nelson (1990) [61] gives the conditions for (Pt,h, σ2t,h)t≥0 to converge weakly to some process
(Pt, σ2t )t≥0 ash→ 0. Suppose that there are constantsξ ≥ 0, θ ∈ R andη > 0 as well as the
starting random variables (P0, σ20) such that (P0h, σ
20h) converges weakly to (P0, σ
20) ash→ 0,
P(σ20 > 0) = 1. Suppose further that
limh→0
h−1ξh = ξ, limh→0
h−1(1− δh − ηh) = 0, limh→0
2h−1η2h = η2. (3.22)
hold, then (Pt,h, σ2t,h)t≥0 converges weakly ash → 0 to the unique solution (Pt, σ
2t )t≥0 of the
diffusion equation
dPt = dµt + σtdWt, t > 0, (3.23)
dσ2t = (ξ − θσ2
t )dt+ ησ2t dBt, t > 0, (3.24)
whereBt andWt are independent Brownian motions.
An example of possible choices satisfying the necessary limit conditions (3.22); ξh = ξh,
δh = 1 − η√
h/2 − θh andηh = η√
h/2. Using these parameters if we rewrite the equation
29
(3.20):
σ2nh = ξh+ η
√h/2
(∆Ptn − ∆nµh)2
h+ (1− η
√h/2− θh)σ2
(n−1)h. (3.25)
When we rewrite the recursive relation for the termσ, we obtain
σ2nh = ξh+
η√
2h(∆Ptn−1 − ∆n−1µh)2 + (1− η
√h/2− θh)(ξh+
η√
2h(∆Ptn−2 − ∆n−2µh)2.
+ (1− η√
h/2− θh)(ξh+η√
2h(∆Ptn−3 − ∆n−3µh)2+
+ (1− η√
h/2− θh)σ2h) · · · ) (3.26)
Rearranging the above equation yields
σ2nh = ξh(1+ (1− η
√h/2+ θh) + · · · + (1− η
√h/2+ θh)i−2)
+η√
2h
n−1∑
j=1
(1− η√
h/2+ θh)i−1− j(∆Pt j − ∆ jµh)2 + (1− η√
h/2+ θh)n−1σ2h. (3.27)
Now we define the iterative threshold function
r In(h) := hαc2(σI
nh)2, (3.28)
whereI ∈ N represents the number of iterations,c ∈ R+ andα ∈ (12,1). c can be regarded
as a constant that determines after the number of standard deviations a price movement is
classified as a jump. Mancini et al. (2010) [?] omit the dependence of thresholdr on h
in their implementation of a similar threshold estimator, since intervals between consecutive
observations of spot market prices are fixed.
(σInh)
2 = ξI h(1+ (1− ηI√
h/2+ θI h) + · · · + (1− ηI√
h/2+ θI h)n−2)
+ηI
√2h
n−1∑
j=1
(1− ηI√
h/2+ θI h)n−1− j(∆Pt j − ∆µh(tn))2
I(∆Pt j−∆µh(t j ))2≤r I−1j
+ (1− ηI√
h/2+ θI h)n−1(σIh)2. (3.29)
By using this filter, we replace the observations with stochastic residual higher than the thresh-
old with the expected value of the Brownian motion, 0. Threshold function satisfies two
necessary asymptotic conditions:
30
1. Ash→ 0, it is obvious thatr I (h) goes to 0 for allI ∈ N.
2. On the other hand, limh→0h log(1/h)
r I (h) = 0 for all I ∈ N if
hαξh(1+ (1− η√
h/2− θh)) + ... + (1− η√
h/2− θh)n−2 + hα η√2h
h log(1/h)+
+(∆Ptn − ∆µh(tn))2 + ... + hα η√
2h(1− η
√h/2− θh)n−2(∆Ptn − ∆µh(tn))2
h log(1/h)+
+hα(1− η
√h/2− θh)n−1σ2
1
h log(1/h)(3.30)
ash → 0. above equation goes to∞. Therefore it can be stated that the threshold function
satisfies the second limit assumption of Theorem 3.2.1.
3.4 Decomposition of the Stochastic Processes
After determining the structural form of the threshold function, the next step is to construct an
algorithm to separate jumps from continuous part. Since the price observations and threshold
function are discrete time processes, the algorithm will be defined on discrete observations.
Moreover this algorithm must both detect the jumps and determine the type of jump process
(whether the jump is a pure jump, spike or semi-spike). Volatility of the diffusion part, jump
processes’ frequency, jump size distributions and mean reversion rateswill be estimated using
the results of this separation.
First of all, definitions which are necessary for this separation processwill be given. The
following framework is taken from the continuous time autoregression (CAR)processes of
Brockwell et al. (2007) [22].
Definition 3.4.1 A CAR(1) process Yt, driven by the Levy processJ(2)
t , t ≥ 0, with parameter
ν ∈ R is defined to be a strictly stationary solution of the stochastic differential equation
dYt = −νYtdt+ dJ(2)t . (3.31)
31
SinceJ(2)t has bounded variation on compact intervals and processYt is Markov due to inde-
pendent increment Levy process, we have for allt > s≥ 0 that:
Yt = e−νtY0 +
∫ t
se−ν(t−u)dJ(2)(u). (3.32)
The necessary and sufficient conditions for the stationarity of the processYt are given in
[21] and [22]. According to these papers, processesYt and Zt, given in (3.5), are strictly
stationary solutions of (3.3) and (3.4) if and only ifν andβ are greater than 0, andY0 and
Z0 have distributions∫ ∞0
e−νudJ(2)(u),∫ ∞0
e−βudJ(3)(u) respectively, given thatY0 andZ0 are
independent ofJ(2)
t , t ≥ 0,J(3)
t , t ≥ 0
respectively andE((J(2)1 )2) < ∞, E((J(3)
1 )2) < ∞.
According to this statement, given thatYt andZt are stationary processes, we conclude thatν
andβ are greater than 0.
Discrete analogous of spike processYnh,n = 0, ..N is represented as the autoregressive pro-
cess (AR(1) process) as follows
Ynh = φY(n−1)h + ∆J(2)tn , (3.33)
whereφ = e−νh. Sinceν > 0, φ ∈ (0,1). From (3.33), we can derive,
∆Yhn = (φ − 1)Y(n−1)h + ∆J(2)tn . (3.34)
The steps followed in parameter estimation are listed below. Also application of these steps
are discussed in the following chapter.
1. Initialization: Using deseasonalized logarithmic price returns we estimate parameters
of GARCH(1,1) process.
2. Detection and Separation of Jump Processes:For the given observation series, it is
assumed that the initial values of processesYt andZt are 0, i.e.,Y0 = 0 andZ0 = 0.
With the estimated GARCH volatility we compute the threshold value. Since jumps are
assumed to be rare events and mean reversion of both spike and semi-spikeprocess are
high, when thenth squared return is higher than threshold level, whole return∆Ptn is
attributed to a jump. Due to fast mean reversion of the jump processes, the cumulative
32
effect of previous jumps are assumed to be contained in the boundaries formedby using
Brownian motion. With a simple calculation fori = 1,2,3;
∆Ptn = ∆J(i)tn ,
∆Ptn+1 = (ρ(i) − 1)∆Ptn + ∆Wtn+1,
... (3.35)
∆Ptn+6 = (ρ(i))5(ρ(i) − 1)∆Ptn + ∆Wtn+6,
whereρ(i) is the corresponding mean reversion rate which is 0 forJ(1).
(a) We take the return higher than the threshold as a jump and replace it with the
expected value of the Brownian component, namely 0.
(b) We form the boundary conditions in order to test whether this jump comes from
a mean reverting process or not. For instance, if there is a positive jump∆J(i)tn for
i = 1,2,3 at nth time interval , and if we assume thatρ(i) ∈ (0,1), for i = 1,2,
boundary condition for the first successive return is defined as
−1× ∆iP− cσi+1 < ∆i+1P < cσi+1. (3.36)
where c is a positive constant. If this condition is satisfied we save this data point,
replace with 0 and recalculate the estimated standard deviation for (n+1)st return.
Implied interval for the mean reversion rate is then calculated as
(∆Ptn+1 − cσn+1
∆Ptn+ 1,∆Ptn+1 + cσn+1
∆Ptn+ 1
)(3.37)
Using this interval and (3.36), corresponding boundary conditions foreach return
is derived.
(c) We examine six returns following each jump point, since most of the jumps are
found to fade away completely until the sixth time interval following the jump.
Successive steps satisfying the boundary conditions are replaced with 0and then
the next return is tested. If the return fails to satisfy the boundary conditions,
algorithm stops and returns to the first step. At every iteration, mean reversion
components for all jumps are retested and reassigned due to changing volatility
estimates. Separated returns are assumed to be the summation of jumps, mean
reversion contributions of former jumps and Brownian increments.
33
3. Classification of Jump ProcessesThe iteration procedure stops when GARCH filter
detects no more jumps. Then, in order to estimate mean reversion rates we classify
collected jump points as pure jumps, spikes and semi-spikes as follows:
• If a jump point is not followed by mean reverting steps, it is classified as a pure
jump.
• If the points separated as mean reversion components jointly satisfies fork =
n, . . . ,n + 6; ||∑k
j=n∆Pt j || − cσk+1 ≤ |∆k+1P|, these points are classified as a re-
alization of spike process. Then the others are classified as semi-spikes.By this
separation, we take jumps with successive observations all of which are inc stan-
dard deviations neighborhood as spikes.
At end of this step, there will be four separate vectors: one vector of pure jumps includ-
ing price steps classified as pure jumps and zeros for the other entries, one vector for
spike process composed of jumps and successive mean reverting returns and zeros; one
vector of semi-spike process observations and one vector of the filteredreturns in which
jumps and mean reverting observations are replaced with zeros while other entries are
equal to returns of deseasonalized price series.
4. Estimation of Mean Reversion Rates:Since we do not have full spike and semi-spike
processes, in order to estimate the mean reversion rates, artificial price series are created
by summing seperated jumps and following mean reverting returns and leaving zeros
unchanged. For illustration we take the spike processY, assuming thatτN(2)(tn) = tn:
PYτN(2)(tn)
=PYtn := ∆Ptn = ∆N(2)(tn)J
(2)
PYtn+1
:=∆Ptn + ∆Ptn+1 = φPYτN(2)(tn)
+
∫ tn+1
tnσsdWs
PYtn+2
:=∆Ptn + ∆Ptn+1 + ∆Ptn+2 = φ2PYτN(2)(tn)
+
∫ tn+2
tn+1
σsdWs (3.38)
Corr(PYτN(2)(tn)
,PYτN(2)(tn)+1) =
E(PYtnPY
tn+1) − E(PY
tn)E(PYtn+1
)√
Var(PYtn)
√Var(PYtn+1)
=
φ
√Var(PY
tn)√
Var(PYtn+1
)
Corr(PYτN(2)(tn)
,PYτN(2)(tn)+2) =
φ2√
Var(PYtn)√
Var(PYtn+2)(3.39)
34
In (3.39), conditional probability of observing another jump in the mean reverting part
is taken to be 0. In addition, the effect of previous jumps are assumed to be indistin-
guishable from the Brownian component. Correlation coefficient estimator, in case of
missing observations proposed by Takeuchi (1995) is
Corr(l) =
∑N−1n=1 PYtnPYtn+l√∑N−1
n=1 a(n+ l)(PYtn)2√∑N−1
n=1 a(n)(PYtn+l)2, (3.40)
wherea(n) = 1 if the nth observation exists, and 0 otherwise. Using (3.40), correlation
of price jumps starting from the first to the sixth successive generated prices are esti-
mated and putting sample variances in (3.39)φ is calculated. After determining mean
reversion rates, we refine price process by subtracting each jumps meanreversion effect
from the rest of the series.
5. Determination and Estimation of Jump Size Distributions: Separated jump obser-
vations in all three jump processes are settled at the tail of the empirical distribution
with a gap around 0. In theory, each tail can be modeled separately. However, since
jumps are rare events, with three different jump processes, it is difficult to find suffi-
cient amount of observations to fit 6 tail distributions. Therefore, absolute value of the
jump point observations are used in specification of the jump size distributions.Among
the long list of jump distributions, the highlighted ones in literature are exponential,
normal, lognormal and inverse Gaussian. Except normal distribution, all of the distri-
butions have positive support. Like Ane et al. (2010), we also add Burrdistribution to
this list and its cumulative distribution function is
F(x) = 1−(1+
(x− µβ
)α−θ, (3.41)
whereα andθ are positive shape parameters,β is positive scale parameter andµ ∈ R is
the location parameter. By taking the absolute value of negative jumps, it is assumed
that both tails are of the same distribution. Empirical weight of each sign among all
jumps is taken as the probability of jumps direction. For each of three jump processes
we estimate the jump distributions. Anderson Darling, Kolmogorov-Smirnov and Chi-
square statistics are used in determination of the jump size distribution. The jump
frequencies are also approximated by the ratio of the number of jump observations of
each process to the total number of observations.
35
6. Estimation of Continuous Process’ Volatility: The last term is the stochastic volatility
of Brownian motion. Although stochastic volatility models, or GARCH type models
can be fitted to the volatility of the filtered price series acquired from Step 4, volatility
is taken constant, not to over parametrize the model.
36
CHAPTER 4
ESTIMATION OF SPOT ELECTRICITY PRICE PROCESS
AND FORECATING SPOT ELECTRICITY PRICES
Parameter estimation of the multi-factor spot electricity price model developed in the previous
chapter will now be carried out in this chapter. Moreover, by using the estimated parameters
day ahead price forecasts are generated and the evaluation of the modelis made. For this
procedure we use four daily average spot price series from Austrian, Italian, Spanish and
Turkish markets. First three countries have relatively mature spot electricity markets, while
liberal Turkish spot electricity market was founded in August 2006 and taken into operation
in December 2009. Austrian market data is taken from EXAA (Energy Exchange Austria),
Austrian energy and environmental exchange founded on 8 June 2001and started spot market
trading in electric power on 21 March 2002. Italian market data is collected from GME, Italian
Power Exchange has been functioning since April 2004. Spanish data istaken from OMEL’s
daily market. For the listed markets equally weighted average of 24 hourly equilibrium prices
constitute the daily observations. Lengths of the observation series are 3399 (from March
2002 to July 2011) for Austria, 2617 (form April 2004 to May 2011) forItaly, 4139 (from
January 2000 to April 2011) for Spain and 577 (from December 2009 toJuly 2011) for Turkey.
In jump process modeling the importance of a long observation series is generally accepted.
Since jumps are assumed to be rare events, probability of acquiring enoughnumber of jump
observations for parameter estimation increases with the length of the series.An infant market
like Turkey may not provide enough data for estimation of our parameter intensive model.
Therefore in order to test the applicability and the validity of our model we usenot only the
series in Turkey but also other three series shown in Figure 4.1.
37
Figure 4.1: Daily average price series
4.1 Model Estimation
Firstly, the weekly pattern in logarithmic prices is extracted according to the procedure de-
scribed in Section 3.1. For all of the price series excluding Turkey, it is observed that this
procedure achieves minor recovery in the weekly autocorrelation. Concluding that the main
problem is due to the chancing magnitude of the weekly cycles through time, we approximate
weekly pattern for each year separately, which helps us overcome the weekly autocorrelation
problem. Due to the relative shortness of the Turkish data, we conclude that one weekly pat-
tern is sufficient to reflect the overall weekly seasonality. Weekly pattern estimated forItaly
is given below. In Figure 4.2 we can see the convergence between the average daily prices
through time. This fact may be explained by developing technology smoothing production
and consumption. After subtracting the weekly cycles, deterministic annual seasonality and
trend function is fitted to the residual, using robust nonlinear fitting algorithm nlinfit, a built
in function in optimization toolbox of MATLAB.
Contrary to some of the similar researches, eliminating jump points before fitting determinis-
38
Figure 4.2: Estimated weekly cycle for Italian electricity market
tic function to avoid bias in estimated parameters, we conclude that results acquired by robust
non linear fitting which uses iteratively reweighted least squares with a bisquare weighting
function are as good as the former method. Coefficients of the deterministic function is given
in Table 4.1 and Table 4.2. For Turkey, the coefficient of the trend parameterα1 has a minus
sign, contrary to an expected positive time effect on prices. Since the day-ahead spot electric-
ity market started full functioning recently in Turkey, this negative trend maybe caused by
falling prices due to increasing efficiency and rising competition with increasing number of
players in the market or falling input prices and demand because of the global financial crisis
affecting the world since 2007.
Applying the threshold function on the residual returns, we acquire the jumppoints and con-
secutive mean reverting returns following the jumps. The parameterc in 3.28 is taken as 2.5
for Austria, 2.5 for Italy, 2.6 for Spain and 2.3 for Turkey and h is assumed to be 1. According
to this filtration the number of jumps are given in Table 4.3. By dividing the numberof jumps
to the total number of observations, the jump intensities are calculated and presented in Table
4.4.
Using Takeschi formula in (3.40), correlation coefficients for 6 lags are computed. Then using
(3.39) mean reversion rates of the filtered spike and semi-spike processes are computed. See
Table 4.5.
Given these autocorrelation coefficients, approximatedν andβ values for Austrian, Italian,
Spanish and Turkish price series are equal to 2.03, 2.41, 1.61, 1.78 (ˆν values), 0.40, 0.40,
39
Mon. Tues. Wed. Thurs. Fri. Sat. Sun.Austria
2002 0.11 0.18 0.18 0.14 0.11 -0.19 -0.532003 0.14 0.21 0.20 0.15 0.08 -0.23 -0.542004 0.08 0.16 0.13 0.10 0.07 -0.15 -0.392005 0.09 0.13 0.14 0.10 0.05 -0.16 -0.342006 0.09 0.16 0.17 0.15 0.08 -0.18 -0.472007 0.11 0.18 0.16 0.16 0.07 -0.21 -0.472008 0.07 0.14 0.13 0.10 0.07 -0.14 -0.382009 0.04 0.12 0.11 0.10 0.05 -0.10 -0.322010 0.06 0.11 0.07 0.07 0.04 -0.09 -0.262011 0.07 0.09 0.08 0.05 0.03 -0.11 -0.21Italy2004 0.07 0.14 0.13 0.180 0.15 -0.17 -0.522005 0.05 0.12 0.13 0.11 0.11 -0.16 -0.352006 0.04 0.08 0.10 0.10 0.07 -0.12 -0.272007 0.04 0.09 0.08 0.07 0.07 -0.09 -0.262008 0.07 0.07 0.07 0.05 0.02 -0.04 -0.162009 0.02 0.02 0.05 0.03 0.02 -0.02 -0.112010 0.01 0.02 0.02 0.03 0.01 0.01 -0.082011 0.01 0.02 0.01 0.01 0.01 0.01 -0.03
Spain2000 0.05 0.08 0.10 0.09 0.11 -0.12 -0.322001 0.07 0.05 0.08 0.09 0.03 -0.06 -0.252002 0.07 0.08 0.06 0.10 0.08 -0.12 -0.272003 0.08 0.06 0.06 0.08 0.09 -0.08 -0.292004 0.04 0.04 0.06 0.050 0.05 -0.05 -0.192005 0.04 0.05 0.08 0.10 0.03 -0.08 -0.222006 0.04 0.05 0.04 0.05 0.02 -0.05 -0.152007 0.04 0.04 0.02 0.03 0.02 -0.03 -0.112008 0.02 0.02 0.01 0.003 0.003 -0.01 -0.042009 0.02 0.01 0.01 0.01 0.02 -0.01 -0.052010 0.04 0.04 0.04 0.04 -0.01 -0.06 -0.102011 0.03 0.03 0.01 -0.001 -0.03 0.01 -0.05
Turkey 0.02 0.07 0.07 0.06 0.04 -0.04 -0.21Table 4.1: Day of the week parameters
Λtn = w(tn) + α0 + α1tn + α2 cos 2π(tn−α3)365
α0 α1 α2 α3
Austria 3.26 0.0002 -0.089 -50.1Italy 4.08 0.0001 0.046 -212.9
Spain 3.37 0.0001 -0.079 -74.5Turkey 4.86 -0.0004 -0.17 254.62
Table 4.2: Deterministic function parameters
40
Figure 4.3: Decomposition of deterministic and stochastic components
Pure Jumps Spike Jumps Semi-spike JumpsAustria 38 100 62
Italy 17 98 39Spain 31 87 53
Turkey 13 23 12Table 4.3: Number of jumps filtered by the iterative GARCH(1,1) volatility threshold r(h)
0.92, 0.43 (β values) respectively. Finding the mean reversion rates for the spike andsemi-
spike process, whole series is refined by subtracting the lagged effects of the previous jumps
from the following observations. And it is observed that this refinement cause minor changes
in the jump sizes as expected. Since spike and semi-spike processes have high mean reversion
rates and jumps are rare enough, effect of the price jumps almost fades away till the occurrence
of the next jump.
After determination of the jump sizes, nognormal, inverse Gaussian, gamma, exponential,
Levy and Burr distributions are fitted to the absolute jump sizes for each of theprocesses.
Then, by dividing the number of observed positive jumps to the total number of jumps, the
41
Figure 4.4: Separated jumps and mean reverting returns
42
Pure Jumps Spike Jumps Semi-spike JumpsAustria 0.011 0.030 0.018
Italy 0.007 0.038 0.015Spain 0.008 0.021 0.013
Turkey 0.023 0.039 0.021Table 4.4: Jump intensities=number of filtered jumps/total number of returns
Austria Italy Spain TurkeySpike process 0.10 0.09 0.20 0.17
Semi-spike process 0.67 0.67 0.40 0.65Table 4.5: Discrete time mean reversion rates
probability of observing a positive jump is derived. For each country we assume that all of the
jumps are coming from the same type of distribution, but only parameters of the distribution
and probability of the jump direction differs. Such a choice is made in order to decrease the
complexity of the estimation and forecasting procedure.
In Table 4.6 estimated parameter values for lognormal and Burr distributions which fit best to
the jumps are given where the lognormal distribution is
F(x) = Φ
(ln(x− γ) − µ
σ
)
and the Burr distribution is
F(x) = 1−(1+
(x− γβ
)α)−k
.
After finding the best fitting distributions to the jumps, we are left with the filtered price
series, assumed to represent the Brownian motion part of the price process. Autocorrelation
functions and the quantile plot of these filtered price returns are depicted inFigure 4.5 and 4.6.
Original series is also presented in order to see the results of our procedure. Although some of
the series like Austria suggest usage of stochastic volatility models in order not to increase the
parametrization, volatility is assumed to be constant. Therefore estimated standard deviations
for the filtered return series are 0.10 for Austria, 0.087 for Italy, 0.11 for Spain and 0.09 for
Turkey.
43
Dist P1 P2 P3 prob(∆J > 0)Austria Lognormal σ µ γ
PJ 1.17 -1.84 0.19 0.47SP 0.67 -1.71 0.14 0.29SS 1.00 -2.07 0.18 0.52
Italy Burr k α β
PJ 0.61 4.09 0.22 0.53SP 0.52 4.63 0.22 0.41SS 0.45 6.02 0.20 0.59
Spain Burr k α β
PJ 0.52 3.85 0.27 0.39SP 0.84 4.21 0.25 0.24SS 0.36 5.36 0.20 0.42
Turkey Burr k α β
PJ 0.35 7.18 0.18 0.31SP 0.58 7.75 0.18 0.35SS 0.29 7.59 0.22 0.67
Table 4.6: Jump distributions
Figure 4.5: Autocorrelation functions of the original return series and thefiltered returns afterthe jump series are extracted
44
Figure 4.6: QQ-plots of the original return series and the filtered returns
4.2 Forecasting
For each of the price series, using the estimated model parameters given in the previous sec-
tion, daily average spot prices are forecasted for the following days. Forecasts are based on
the paths created by Monte Carlo simulation. Forecast periods are 91 daysfor Austria (from
July 11 to October 9), 119 days for Italy (from May 29 to September 24), 154 days for Spain
(from April 26 to September 26) and 91 days for Turkey (from July 1 to September 29). The
forecasts are made weekly using the available data and model parameters, 100000 paths are
generated for the following seven days. Mean value of the generated daily values are taken as
forecasted price of the corresponding day. Then the realized price for that week is added to
the observation series and the whole procedure is repeated beginning from the jump detection
step. By using the newly created jump series, forecasts for the following week are generated.
Original price series starting from January 1st, 2011 to the end of the forecasting period and
zoomed the realizations and forecasted prices in the corresponding forecast period are given
in Figure 4.7 and Figure 4.8.
In order to check forecast accuracy, daily analougus of linear MeanWeekly Error (MWE)
45
Figure 4.7: Observed and forecasted price series starting from 2011
Figure 4.8: Observed and forecasted price series through the forecast period
46
given in [78] for hourly data is computed for each week:
MWE=17
7∑
d=1
∣∣∣Sd − Sd
∣∣∣S7
,
whereS7 is the mean observed price for the given week. MWE values for each week and
their mean value are given in Figure 4.9.
Figure 4.9: MWE of each week in the forecast period and their average
According to MWE’s of each week and their corresponding mean levels, itcan be concluded
that estimated model shows the best performance for the Spanish data then Italy, Turkey and
Austria follow in order.
47
CHAPTER 5
ELECTRICITY FUTURES MARKETS
In deregulated electricity markets, it is generally observed that increasingcompetition in retail
electricity markets results in greater price volatility as the industry encouragesmarket driven
prices by moving away from administratively determined, cost-based rates and encourages
market-driven prices. Price volatility introduces new risks for generators, consumers, and
marketers. In a competitive environment, some generators will sell their power in potentially
volatile spot markets and will be at risk if spot prices are insufficient to cover generation costs.
Consumers will face greater seasonal, daily, and hourly price variability and, for commercial
businesses, this uncertainty could make it more difficult to assess their long-term financial po-
sition. Finally, power marketers sell electricity to both wholesale and retail consumers, often
at fixed prices. Marketers who buy on the spot market will face the risk that the spot market
price can substantially exceed fixed prices specified in contracts.
Electricity futures and other derivatives can help each of these market participants to man-
age, or hedge, price risks in a competitive electricity market. Futures contracts are legally
binding and negotiable contracts that call for the future delivery of a commodity. In most
cases, physical delivery does not take place, and the futures contract is closed by buying or
selling a futures contract on or near the delivery date, or by design, thecontracts are subject to
financial settlement. While the futures contracts are traded in organized exchanges, forwards
and various types of options are traded over the counter (OTC). The first exchange to introduce
electricity futures was the Scandinavian power exchange Nordpool, where monthly, quarterly
and annual futures contracts are traded. Besides the New York Mercantile Exchange, London
International Petroleum Exchange, UK Power Exchange, European Power Exchange, Pow-
ernext, APX UK are all well known exchanges where electricity futures are traded. Most of
48
the national deregulated spot electricity markets around the world are completed with deriva-
tive exchanges where trading of standardized electricity contracts trading take place.
On the other hand, a large variety of electricity derivatives are traded among market par-
ticipants in the OTC markets, including forward contracts, swaps, plain vanillaoptions and
exotic options like spark spread options, swing options, swaptions and so on. In this chapter,
only the standardized futures contracts will be discussed. A detailed discussion of other in-
struments can be found in [33]. The main reason of concentration on the futures markets is to
gain an insight about the monthly electricity futures contracts which have been traded since
September 2011 in the Turkish Derivatives Exchange (TurkDEX). Moreover, it is widely ac-
cepted that futures being traded on organized exchanges reflect a higher market consensus and
transparency than other OTC traded products. In addition, credit risk and monitoring costs
constitute a lower part of the futures prices than the other products since exchanges implement
strict margin requirements to ensure financial performance of all trading parties.
A typical futures contract is a standardized, transferable and obligatory contract to buy or
sell a specified quantity of the underlying asset at a particular future point in time (maturity)
for a specified price contracted today (futures price). The seller of thecontract is also obliged
to sell the underlying asset, in our case the electricity. The maturity, quality andquantity of
the underlying asset are all standardized. The only negotiable aspect of the contract is the
fixed price paid for the underlying asset at maturity; the futures price. There are no initial
costs of entering a futures contract. Due to changing market conditions and price expecta-
tions, the value of a particular futures contract, however, does changeover time. The value
of each futures contract is marked to market according to the calculated market value for that
trading day. This means that financial positions are valued based on the current fair market
price. Differences between previous day’s value and the current market price are settled im-
mediately and the gain/loss of a position is added/withdrawn from the margin account of the
position holder. Since the risk for both parties is unlimited theoretically, exchanges use these
margin accounts to guarantee that the contract obligations would be fulfilled.
Different from other future contracts, electricity futures have a delivery period instead of a
fixed delivery time. Generally contracts with weekly, monthly, quarterly and annual delivery
periods are traded. The regulations, number of traded contracts or trading and delivery peri-
49
ods varies across the countries. For instance, in Turkey only the monthly base load contracts
are traded at the moment. The reference price for TurkDEX base load electricity futures con-
tracts are the average of the day-ahead hourly prices of the corresponding delivery month.
Contract size is equal to the number of hours in the contract month times 0,01 MWh. The
quoted futures price is the price of 1 MWh. Contracts for the current monthand the following
three months are traded simultaneously. Contracts are traded until the last trading day of the
delivery month. Daily settlement price is the weighted average price of all transactions per-
formed within the last 10 minutes before the closing of the trading session.
There is a growing literature on pricing of electricity derivatives. In the following section,
future contract pricing models proposed in literature will be examined. However, since al-
most all of these models need a long stream of observations for parameter estimation, in
Section 5.2 an alternative methodology which can be used in an infant derivative market like
Turkey’s is discussed. Although most of the models described in this chapter are inapplicable
for the market in Turkey due to the lack of historical data, they are discussed in detail since
the development and calibration of such a stochastic model will be one of the further research
topics.
5.1 Stochastic Futures Pricing Models in Literature
Parallel to the increasing importance of derivative contracts in the power markets, researches
aiming to model these contracts have also been increasing. However, sincethe electricity is
not a tradable asset in the classical sense due to its non-storability, models used in financial
markets need some modifications. In the literature, electricity futures price process modeling
has been done by following either the spot price approach or the futuresprice approach. In
the spot price approach, firstly electricity spot price process is accurately modeled and then
by using additional conditions which are related to spot and futures prices, futures price pro-
cess is derived. In the futures price approach, instead of modeling the spot price and deriving
futures prices, the futures prices are directly modeled. This approach isbased on Heath-
Jarrow-Merton (HJM) framework which is developed for fixed income markets. Moreover,
there exists two types of applications for the HJM framework in electricity futures pricing.
The first approach initially models the futures contracts with fixed delivery and then corre-
50
sponding dynamics for futures, delivering electricity during a period arederived. The second
approach is directly modeling the futures price with a delivery period (For asummary of
HJM framework Appendix A.2). These three approaches (spot price approach, futures price
approach based on fixed delivery contracts, futures price approach based on futures contracts
with delivery periods) and related literature are summarized in the following sub-sections.
In this context, as a result of a general literature review, widely acceptedfeatures of the fu-
ture contract prices are listed below. Any futures contract model is expected to contain below
mentioned features:
• Future contracts show lower price volatility than the spot asset. Short term changes
or price shocks have less effect on the futures prices, since futures prices depend on
the average spot price observed during the delivery period and most of the spot price
shocks are not persistent as it is shown in the previous chapter.
• Length of the delivery period affects the futures price volatility. The longer the delivery
period, the lower the futures price volatility.
• Time left to delivery is also proved to be effective on the futures price volatility. This
effect is known as the Samuelson effect, as the maturity approaches volatility observed
in the futures price also rises.
• Futures prices are also found to exhibit a seasonal pattern like the spot electricity prices.
For instance, if the spot electricity prices are higher in the winter months, futures con-
tracts with winter delivery are also expected to have higher prices.
5.1.1 Spot price approach
Main stream spot electricity price models used in the literature have already been discussed in
the previous chapter. In this context, Lucia et al. (2002) calibrate their one and two factor spot
price models to derivative contracts. Pilipovic (1998) uses a two factor spot electricity model,
which leads to a complicated closed form expression for fixed delivery futures. Benth et al.
(2008b) calculate the fixed delivery forward contract prices by usingtheir multi-factor expo-
nential spot electricity price models. However, as it is stated in Benth et al. (2009a) all of these
51
models exhibit market incompleteness due to jump processes in the models and/or mutlidime-
sional Brownian motions driving the spot price dynamics. Moreover, the non-storability of
the underlying commodity rules out the possibility to hedge the derivatives by trading the
underlying. In this context any probability measureQ equivalent to the objective probability
measureP is risk neutral and due to the absence of perfect hedge, market price of risk has to
be estimated.
For a given spot price which is represented with semimartingaleSt and constant risk free
interest rater, under the risk neutral probability measureQ discounted expected payoff of the
futures contact must be equal to its price today. Since it is costless to enter these contracts
e−r(T−t)EQ(ST − f (t,T)|Ft) = 0, (5.1)
where f (t,T) is the price of futures contracted started on dayt with delivery on dayT, t ≤ T.
Assuming thatST ∈ L1(Q), the space of integrable random variables with respect toQ and
f (·,T) is an adapted process
f (t,T) = EQ(ST |Ft). (5.2)
With the same line of reasoning and the assumption that the settlement takes place continu-
ously on the delivery period, for the futures contract with delivery period [T1,T2], we obtain
EQ
(∫ T2
T1
e−r(u−t)(Su − F(t,T1,T2))du|Ft
)= 0, (5.3)
F(t,T1,T2) = EQ
(∫ T2
T1
re−ru
e−rT1 − e−rT2Sudu|Ft
). (5.4)
Using the Fubini theorem, we get
F(t,T1,T2) =∫ T2
T1
re−ru
e−rT1 − e−rT2f (t,u)du. (5.5)
If the settlement takes place at the end of the delivery period,T2,
F(t,T1,T2) =∫ T2
T1
1T2 − T1
f (t,u)du. (5.6)
As it is said at the beginning, determination of the risk free probability measureis the cru-
cial step in futures contract pricing, using the spot price approach. For instance in Benth et
al. (2008a) the Esscher transform is used in order to restrict the possible set of equivalent
martingale measures. However, in most cases it is not possible to find an analytic solution
for F(t,T1,T2) processes. In general, multi-factor spot electricity models aiming a detailed
52
description of the spot prices lead to complicated future price dynamics. As aresult, not only
the calibration of the pricing measure on the market data becomes a very challenging task
in the electricity markets, but also the flexibility of the spot models are rarely sufficient to
generate future curves that are consistent with the observed curves.
5.1.2 Instantaneous delivery models
In fixed income markets, instead of modeling the prices, the forward rates are directly mod-
eled using the HJM approach. In analogy to HJM approach for the forward interest rates,
many authors have proposed a HJM type model for electricity forward andfuture curves. For
instance, Koekebakker et al. (2005), construct a financial market where the uncertainty is de-
scribed by aK dimensional Brownian motion (W1, ...,Wk), which is defined on the probability
space (Ω, F,Q) with the filtrationF satisfying the usual conditions. The probability measure
Q represents the equivalent martingale measure and risk free rate,r, which is assumed to be
constant. Assuming that the futures market is represented by a continuousfutures price func-
tion f (t,T), where the futures price processes are martingales underQ by construction. Two
types of models are proposed in Koekebakker et al. (2005);
• Futures price process which is independent of the futures price level:The dynamics are
given by
d f(t,T) =K∑
i=1
σAi (t,T)dWi(t), (5.7)
where (W1, ...,Wk) are independent Brownian motions andσAi (t,T) are time dependent
deterministic volatility functions. The solution of the (5.7) and the distribution of the
futures prices are
f (t,T) = f (0,T) +K∑
i=1
∫ t
0σA
i (s,T)dWi(s), (5.8)
f (t,T) ˜ N
f (0,T),K∑
i=1
∫ t
0(σA
i (s,T))2ds
. (5.9)
• Futures price process which is proportional to the futures price level:The dynamics of
53
the futures price are given by
d f(t,T)f (t,T)
=
K∑
i=1
σBi (t,T)dWi(t), (5.10)
the solution of (5.10) and the distribution of the futures prices are
f (t,T) = f (0,T)exp
−12
K∑
i=1
∫ t
0(σB
i (s,T))2ds+K∑
i=1
∫ t
0σB
i (s,T)dWi(s)
, (5.11)
ln f (t,T) ˜ N( f (0,T) − 12
K∑
i=1
∫ t
0(σB
i (s,T))2ds,K∑
i=1
∫ t
0.(σB
i (s,T))2ds) (5.12)
Since the HJM framework does not imply a specific model for the spot electricity prices
and the volatility functions used in the above equations are flexible enough, awide variety
of future price dynamics can be constructed from the above models. For instance, one factor
spot electricity price model proposed by Luica et al. (2002) is consistentwith the futures price
model in (5.8) takingσA(t,T) = σe−κ(T−t) whereσ andκ are positive constants. Bjerksund et
al. (2000) propose two different models for fixed delivery futures price modeling. Their one
factor model has the volatility functionσB(t,T) = aT−t+b + c, wherea, b andc are positive
constants. For both of the Lucia et al. (2002) and Bjerksund et al. (2000) volatility functions
decrease with maturity and approach to 0 asT → ∞. Moreover Bjerksund et al. (2000)
propose a three factor model with volatilitiesσB1 (t,T) = a
T−t+b, σB2 (t,T) = ( 2ac
T−t+b)12 and
σB2 (t,T) = c where all parameters are assumed to be positive. In their paper authors argue
that the one factor model is adequate for contingent claim pricing, while three factor model
has better performance in risk management. ThenF(t,T1,T2) being today’s contract price
with delivery period [T1,T2] wheret ≤ T1 < T2, and assuming that the contract price is paid
as a constant cash flow during the delivery period; the price of this contract is given as
F(t,T1,T2) =∫ T2
T1
w(r,u) f (t,u)du, (5.13)
where
w(r,u) =e−r(u−t)
∫ T2
T1e−r(u−t)du
. (5.14)
Although by using (5.13) and (5.14), dynamics of the actually traded futures can be captured,
as Benth et al. (2008b) argue that with this method, the implied dynamics of the futures
contract price with delivery period [T1,T2] can become very complicated. In their referenced
54
paper, dynamics are examined by using the fixed time delivery futures contract following the
stochastic differential equation presented in (5.15):
d f(t,T) = σ(t,T) f (t,T)dW(t). (5.15)
Then the implied dynamics for the futures contract with delivery period [T1,T2] is given as;
dF(t,T1,T2) = Σ(t,T1,T2)dW(t), (5.16)
where the volatility dynamics of the futures contract with delivery period [T1,T2] is
Σ(t,T1,T2) =∫ T2
T1
w(u,T1,T2)σ(t,u) f (t,u)du (5.17)
and
w(t,T1,T2) :=e(−rt)
∫ T2
T1e(−ru)du
(5.18)
for the given constant risk free interest rater. After integration by parts we see
Σ(t,T1,T2) = σ(t,T2)F(t,T1,T2) −∫ T2
T1
δ2σ(t,u)∫ u
T1
w(τ,T1,T2) f (t, τ)dτdu. (5.19)
In (5.19)δ2 denotes partial differentiation with respect to the second variable of the respective
function. Since ˆw(τ,T1,T2)/w(τ,T1,u) is independent ofτ,
Σ(t,T1,T2) = σ(t,T2)F(t,T1,T2) −∫ T2
T1
δ2σ(t,u)w(τ,T1,T2)w(τ,T1,u)
F(t,T1,u)du (5.20)
Then using (5.16) and (5.20) yield
dF(t,T1,T2) = σ(t,T2)F(t,T1,T2)dW(t) −∫ T1
T2
δ2σ(t,u)w(τ,T1,T2)w(τ,T1,u)
F(t,T1,u)dudW(t).
(5.21)
In Benth et al. (2008b) it is shown that when the volatility functionσ is not a function of
the expiration date of the contract, i.e.,δ2σ(t,u) = 0, F(t,T1,T2) has lognormal dynamics.
However, it is emphasized that in realistic models the volatility depends strongly on the time
55
of delivery of the contract. Moreover, if we start with a model for futurecontracts with
fixed delivery, we need to estimate the parameters using the market traded future contracts
with delivery over a period. Due to the possibility of ending with complicated dynamics for
the futures price process, it may not always be possible to integrateF(·,T1,T2) in order to
estimate the model parameters.
Alternatively, instead of fitting the parameters of the observed futures price processF(·,T1,T2),
f (·,T) prices can be extracted from the observed futures contract prices.This can be done by
using the smoothing technique proposed by Benth et al. (2007b). Then, the theoreticalf (·,T)
processes constructed under the risk neutral probabilityQ are transformed according to the
objective probability measureP and the parameters are estimated.
5.1.3 Direct modeling of the observed future contracts
Since the instantaneous delivery future contracts are not actually tradedin the electricity mar-
kets, in order to avoid additional data formation problem mentioned in the previous subsection
and model complications that may arise, Benth et al. (2008b) propose direct modeling of the
future contracts with the delivery period [T1,T2], which are the contracts traded in organized
exchanges. In this context, authors adapted the HJM framework and theyspecify arbitrage
free dynamics for the processF(·,T1,T2), which are valid for all delivery periods within a
predetermined time horizon. However, the authors concluded that this goalis hard to achieve
while preserving the flexibility of the models which can be easily adapted to optionpricing
and risk management models. Moreover, it is shown that a lognormal model for F(·,T1,T2)
dynamics cannot satisfy the no arbitrage condition and also has a volatility process depend-
ing on the delivery period of the contract at the same time. The solution to this problem is
given as modeling not all of the available contracts simultaneously but modelingthe building
blocks of the futures market. This means modeling the contracts that cannot be decomposed
into other traded contracts. For instance, if 1 year contract and 12 monthscontracts for each
month of the following year are traded in the market simultaneously,F(·,T1,T2) process only
models the monthly contracts and ignores the yearly contract. In this framework, a lognormal
one factor future price dynamics under the equivalent risk neutral martingale measureQ is
given by
dF(t,T1,T2) = Σ(t,T1,T2)F(t,T1,T2)dW(t), (5.22)
56
whereΣ(t,T1,T2) is a continuously differentiable and positive function representing the volatil-
ity and W(t) is a Wiener process under the risk neutral measure. When it is assumed that the
settlement of the contract takes place at the maturityT2, market futures volatilityΣ(t,T1,T2)
can be associated to the instantenous delivery futures volatilityσ(t,T) as follows:
Σ(t,T1,T2) =1
T2 − T1
∫ T2
T1
σ(t,u)du. (5.23)
Considering the (5.23), Benth et al. (2008b) examine six different volatility functions, which
are suggested and used in the commodity market researches. And they testtheir performances
in modeling futures price process,F(·,T1,T2). First of the examined volatility function be-
longs to the Schwartz’s (1997) one factor oil price model;σ(t,u) = ae−b(t−u), where a,b
≥ 0. By the (5.23) corresponding volatility is found asΣ(t,T1,T2) = aφ(T1,T2), where
φ(T1,T2) = e−b(T1−t)−e−b(T2−t)
b(T1−T2) . The basic constant volatility model appears ifb = 0. While
the Schwartz model reflects only the maturity effect and ignores the seasonality, the instante-
nous delivery futures volatilityσ(t,u) = a(t)e−b(u−t) picks up also the seasonality effect with
the terma(t), wherea(t) = a +∑J
j=1(d j sin(2π jt) − f j cos(2π jt)). In this model t is given in
years,d j and f j are real constants anda,b ≥ 0. ThenΣ(t,T1,T2) = aφ(T1,T2). The fourth
instantenous volatility isσ(t,u) = a((1 − c)e−b(t−u) + c), wherea,b ≥ 0 and 0≤ c ≤ 1.
The associated volatility forF(t,T1,T2) is Σ(t,T1,T2) = a((1 − c)φ(T1,T2) + c), where
φ(T1,T2) = e−b(T1−t)−e−b(T2−t)
b(T1−T2) again. When this model is combined with seasonal spot volatility,
σ(t,u) = a(t)((1−c)e−b(t−u)+c) then futures volatility isΣ(t,T1,T2) = a(t)((1−c)φ(T1,T2)+c),
where a,b ≥ 0, 0 ≤ c ≤ 1 d j and f j are constants. The last model isΣ(t,T1,T2) =
a(t) + cφ(T1,T2) with σ(t,u) = ce−b(t−u) + a(t), wherea(t) modeling the seasonality as it
is given before. According to this model the spot price volatility (σ(t, t)) is a(t) + c and long-
run volatility (u→ ∞) is governed by the seasonality terma(t). Authors note that with these
properties, last model has a clear separation of maturity and seasonal effects.
It is assumed that under the physical probability measureP, dynamics of financial electricity
contracts traded at exchanges can be described as
dF(t,T1,T2) = λΘ(t,T1,T2)F(t,T1,T2)dt+ Θ(t,T1,T2)F(t,T1,T2)dW(t), (5.24)
whereB(t) is a Brownian motion underP, λ represents the market price of risk (which is
assumed to be constant) and the functionΘ(t,T1,T2) is deterministic. Then the logarithmic
returns of the futures contract (r(t,T1,T2)) is equal to;
57
ln(F(t + ∆t,T1,T2)
F(t,T1,T2)) =
∫ t+∆t
t(λΘ(s,T1,T2) − 1
2Θ2(s,T1,T2))ds
∫ t+∆t
tΘ(s,T1,T2)dB(s). (5.25)
Therefore logarithmic returnsr(t,T1,T2) are normally distributed with meanm(t,T1,T2) =∫ t+∆t
t(λΘ(s,T1,T2)− 1
2Θ2(s,T1,T2))dsand variancev(t,T1,T2) =
∫ t+∆t
tΘ2(s,T1,T2)ds. Then
the return of the ith contract is
r i(t,T1,T2) = mi(t,T1,T2) + vi(t,T1,T2)ǫ i(t), (5.26)
whereǫ(t) ˜ N(0,1). In Benth et al. (2008b) parameters are estimated using log-likelihood
functions and it is concluded that maturity effect is very significant and modeling this effect
with a simple exponential function is insufficient. Estimation results also verify the existence
of seasonal volatility, and an additive specification is able capture the maturityand seasonality
effects in the volatility.
5.2 Risk Premium Approach
Direct modeling of futures price processes approach, discussed in theprevious section, uses
future contracts price observations for the parameter estimation. However, in Turkish elec-
tricity markets, due to the lack of historical futures market price data, we have to build our
own methodology on the spot electricity prices. In Subsection 5.1.1, we see that modeling
the electricity futures price process by using the spot price process requires identification of
risk neutral probability measureQ. Researchers usually connect the risk neutral probability
to the concept of market price of risk. The market price of risk is the difference between the
drift in the original probability measureP and the drift in the risk neutral measureQ in the
stochastic differential equation governing the spot price dynamics and it is assumed to reflect
how investors are compensated for bearing risk by holding the asset.
Another important and widely referred quantity relating futures and expected spot prices is
the market risk premiumπ(t, (T2 − T1)). This premium is defined as the difference of the
electricity futures priceF(t,T1,T2) and the conditional expectation of the average day-ahead
price of electricity during the future delivery period [T1,T2], with respect to the objective
probability measureP. It is assumed to depend on time to maturity and length of the delivery
58
period:
F(t,T1,T2) = EP
[1
T2 − T1
∫ T2
T1
Sudu|Ft
]+ π(t, (T2 − T1)). (5.27)
According to the Keynesian theory, market risk premium depends on the risk preference of
the hedgers and speculators. Therefore, the future curves observed in the market reflect not
only the forecasts of the commodity spot price in the future but also the dominant hedging
tendency in the market. Assuming that the main motivation for the agents to engagein the
future contracts is risk diversification, producers, who have made substantial investments,
have incentive to reduce variability in their profits by trading these instruments. Similarly,
consumers (can be either intermediaries or use electricity in their production process) also
have incentive to hedge their positions in the market and to diversify the market price risk.
The hedging and risk diversification preferences of producers andconsumers generally cover
different time horizons. For instance, while a producer is exposed to market uncertainty for
a longer period of time, determined by the remaining life of its assets, a consumerhas to
make decisions for shorter periods. In other words, gains in terms of riskdiversification for
consumers and producers vary across time. These differences in the hedging preferences and
imbalances between buyers and sellers of the future contracts are assumed to create the mar-
ket risk premium. In Benth et al. (2006) it is stated that the further out one looks, consumers
will have less incentives to buy future contracts where producers’ desire to hedge does not di-
minish as quickly as consumers’. Situations whereπ(t, (T2−T1) > 0 (positive market risk pre-
mium) are associated with higher consumer demand to cover their positions than producers.
Conversely, when the producers’ desire to hedge outweighs consumers’, π(t, (T2 − T1)) < 0
(negative market risk premium).
After calibration of the spot electricity model, according to (5.27), risk premium must be
added in order to price the derivative instrument. In estimation of the risk premium, ex-ante
and ex-post estimation methods are used in the literature. Ex-ante or expected risk premium
method uses the original definition of the risk premium given in (5.27). In this methodology
the choice of an appropriate spot price model is essential for the derivation of a consistent risk
premium. Claiming that by using ex-ante method it is difficult to reach consistent and robust
results, some of the researchers propose an ex-post approach. The ex-post risk premium is
given as
F(t,T1,T2) =∫ T2
T1
Sudu+ π(t, (T2 − T1)). (5.28)
59
In this approach, instead of the expected average day ahead prices, the average of the observed
market prices through the delivery period [T1,T2] are used. Moreover, subtracting∫ T2
T1Sudu
from both sides of (5.27) gives
F(t,T1,T2) −∫ T2
T1
Sudu= EP
[1
T2 − T1
∫ T2
T1
Sudu|Ft
]−
∫ T2
T1
Sudu+ π(t, (T2 − T1)),
(5.29)
F(t,T1,T2) −∫ T2
T1
Sudu= π(t, (T2 − T1)) + ǫt. (5.30)
Therefore, under the assumption that market participants form their forecasts based on rational
expectations the ex-post risk premium equals the ex-ante risk premium plus anoise term
with a mean equals to 0. By subtracting average realized spot prices duringthe delivery
periods from historically observed futures prices and assuming that trader expectations are
unbiased in the long run, risk premiums are studied empirically. This approachis applied
by Wilkens et al. (2007) who examine futures prices on the German EEX market. They
find positive but highly volatile risk premiums for futures contracts with times to maturity up
to six months. In the Nordic electricity markets, Botterud et al. (2002) identify positive risk
premiums for futures contracts with a time to maturity up to one year. On the other hand, many
researchers have modeled the extracted risk premium series. Bessembinder et al. (2002) study
the electricity forwards and concluded that the forward risk premium is negatively related to
the variance and positively related to the skewness of expected electricity spot prices. Douglas
et al. (2008) relate observed risk premiums to indirect storability by showingthat higher
natural gas inventory levels reduce the forward risk premium in the PJM market, especially
during extremely warm and cold periods.
As it is already mentioned, for now it is not possible to discover the appropriate dynamics
governing neither for the futures price nor for the related risk premium in Turkish electricity
futures market. Although it is possible to determine the necessary building blocks, without
calibration and testing the significance of the theoretical factors, applicabilityof any model
will be limited. However, we can still propose an elementary approach, in order to decide
whether or not to buy/sell a given futures contact as long as we have a sound model for the
spot electricity model by using (5.27). For a given future contract’s price,F(t,T1,T2), first of
all the expected average spot price for period [T1,T2], EP
[1
T2−T1
∫ T2
T1Sudu|Ft
]is found using
a Monte Carlo simulation. Future contract prices higher than the expected average spot price
(F(t,T1,T2) > EP
[1
T2−T1
∫ T2
T1Sudu|Ft
]), offers a positive risk premium and can be used by the
60
Figure 5.1: TURDEX-December 2011 contract
producers or any investor who wants to take a short position on these contracts. Contrarily,
whenF(t,T1,T2) < EP
[1
T2−T1
∫ T2
T1Sudu|Ft
], future contract has a negative risk premium and
is appropriate for taking a long position. As a result, 0 risk premium appearsas a natural
boundary for a risk neutral investor.
In order to test our simple approach, we will use the only contract that hasbeen traded in TUR-
DEX since 26 September 2011. This contract is a base load contract with delivery through
December 2011. It is obvious that only one contract is not sufficient to reach robust conclu-
sions. However, considering that this illiquid and thin market conditions will continue for a
while, we can still see whether our approach leads to positive pay off or not. Development of
the daily settlement price and number of transactions for December 2011 base load contract
is shown in Figure 5.1.
Starting with the first trading day of the futures contract (26 September 2011), for every Mon-
day from 26 September to 26 December we form monthly average expected spot electricity
price for December, by using our model given in chapter 3 (5000 Monte Carlo simulations).
Taking the future contract price at that day as given, we calculate the risk premium for the
contract as equal toF(·,T1,T2) − EP[ 1T2−T1
∫ T2
T1Sudu|F·]. Theoretically if this premium is
different than zero, either long or short position holders are expected to gain a positive risk
premium. Therefore, if the mean value of the simulated risk premium is higher thanzero,
61
we conclude that taking a short position can yield a positive return. If the risk premium is
negative, we conclude that taking a long position is appropriate. And we compute the revenue
of holding the suggested position in the futures contact, due to the daily mark to market. First
of all, simulated risk premium distributions for four days of each month are given in Figure
5.2.
Figure 5.2: Spot electricity prices
From the risk premiums in the Figure 5.2, except 26 December, our naive approach leads us to
long positions. The dates, direction of the estimated risk premium and the suggested positions
are given in Table 5.1).
Lastly as of January 1st, 2012, calculated daily pay offs of the suggested positions for the
given dates which are shown in Figure 5.3. Considering each December contract is written
on 744 MWh electricity, 11 out of 13 positions taken according to our approach ended the
contract with a positive payoff. It is far out of reach to conclude that, suggested elemen-
tary approach is an alternative for the market agents. However, as a promising method, we
continue to analyze and test this method.
62
Date Risk Premium Sign Suggested Position26 September - long
3 October - long10 October - long17 October + short24 October - long31 October - long
14 November - long21 November - long28 November - long5 December - long12 December + short19 December + short26 December + short
Table 5.1: Suggested future contract position according to the risk premium
Figure 5.3: Suggested future positions’ pay offs
63
CHAPTER 6
CONCLUSION AND OUTLOOK
In the past two decades, electricity industries in many countries which were initially designed
as vertically integrated national or state dominated monopolies have been experiencing a
deregulation process. The increase in the number of market players together with the com-
petition and development of relatively more liberal electricity markets caused electricity to
become a commodity whose price is determined by supply and demand. Day-ahead spot
electricity markets, are the most transparent spot markets where one can find integrated sup-
ply and demand curves of the market players for each settlement period. We model spot
electricity prices, since it is an indicator for the market players and regulators. Logarithmic
daily average spot electricity prices are modeled as a summation of a deterministicfunction
and multi-factor stochastic process. Randomness in the spot prices is assumed to be governed
by pure jumps and mean reverting jump processes additional to a Brownian motion.
In order to estimate the model parameters, following Mancini’s (2009) [54] approach, jump
processes are seperated by using a parametric threshold function whichis composed of a
multiple of stochastic volatility estimate generated by GARCH(1,1) model. Although the
idea of using a threshold function for the separation of jumps is not original,using a GARCH
type threshold in electricity price modeling is uncommon. By including two mean reverting
processes instead of one, we can separate price jumps being effective only for one day which
are mostly due to hourly jumps in any given day from the jumps that affect the price level for
more than a day. One of the main goals of this thesis is to propose spot and future contract
price models which can be used in recently established liberal Turkish electricity spot market
and electricity future contracts that are traded in the national derivativesexchange. However,
since we do not prefer to be restrained by the small sample size, we test ourmodel with
relatively more mature markets’ spot electricity data. For all of the examined countries, the
64
separation results are found to be in accordance with our initial expectations. The occurrence
of pure jumps and semi-spikes are less than price spikes as expected, andjumps are rare events
as it is confirmed by their low jump intensity estimations. Burr distribution is also found to
be good at capturing the distributional properties of the electricity price jumpsbesides the
widely excepted jump distributions. Moreover, the week ahead forecast performance of the
model shows that GARCH threshold multi-factor jump model can also be a useful alternative
for the market practitioners.
In the derivatives front, although we summarize three main approaches used for electricity
forward and future contracts modeling in the literature, we cannot propose a contract price
model due to data shoratge. Instead of proposing a future contracts price model, we offer a
decision technique where the given contract prices are used. With this technique which is built
on the risk premium theory, derivative market players can decide whether to take a long or a
short position. After testing our technique, we conclude that the decision rule is promising
but needs more empirical research.
By taking this thesis as a starting point in electricity market modeling, further research can
develop hourly spot electricity model and use this model in pricing future contract valuation
by defining a new risk measure which can be applied to electricity portfolios. Since hourly
day-ahead prices are more likely to be 24 different series than a single hourly price series,
panal data techniques are assumed to be employed.
65
APPENDIX A
Preliminaries
A.1 Definitions
Definition A.1.1 A cadlag, adapted, real valued stochastic process L= (Lt)0≤t≤T with L0 = 0
a.s. is called a Levy process if the following conditions are satisfied:
1. L has independent increments, i.e., Lt − Ls is independent of Fs for any0 ≤ s< t ≤ T.
2. L has stationary increments, i.e., for any0 ≤ s< t ≤ T the distribution of Lt+s−Lt does
not depend on t.
3. L is stochastically continuous, i.e., for every0 ≤ t ≤ T andǫ > 0, lims→t P(|Lt − Ls| >
ǫ) = 0.
A Levy processXt which has a characteristic triplet (γ,A, ν) and therefore a characteristic
function
E[eiuXt ] = exp
(iνu− 1
2u2A+
∫ ∞
−∞(eiux − 1− iuI |x|≤1)ν(dx)
)(A.1)
is a finite activity jump process ifν(R) < ∞, i.e., almost all paths ofXt have finite number of
jumps on every compact interval.
A.2 Heath Jarrow Merton Framework for the Stochastic Modeling of Interest
Rate Dynamics
Related concepts are listed below
66
• The short rate,r(t), is the annualized interest rate at which an entity can borrow money
for an infinitesimally short period of time from time .
• Instantaneous forward rate,f (t,T) is the annualized interest rate, contracted att, over
the infinitesimal small interval (T,T + ∆t).
• A zero-coupon bond with maturity dateT is a contract that guarantees 1 unit of payment
on the date T. The price at timet of a zero-coupon bond with maturityT is denoted by
P(t,T).
Then
f (t,T) := −d logP(t,T)dt
, (A.2)
r(t) = f (t, t), (A.3)
and
P(t,T) = exp
(−
∫ T
tf (t, s)ds
). (A.4)
In the HJM framework, it is assumed that under the equivalent martingale measureQ
d f(t,T) = α(t,T)dt+ σ(t,T)dW(t), (A.5)
f (0,T) = f ∗(0,T),
whereW is a d-dimensional Brownian motion underQ, α represents the drift of the forward
rates,σ represents the volatility of the forward rates andf ∗ (0,T) is the observed initial
forward curve.
Since the forward rate dynamics are modeled directly under the martingale measureQ, prices
are arbitrage free. Then market satisfies the following equations:
P(0,T) = exp
(−
∫ T
0f (0, s)ds
), (A.6)
P(0,T) = EQ[exp
(−
∫ T
0r(s)ds
)], (A.7)
if the HJM drift condition holds.
HJM drift condition: Assume that the family of forward rates is given by Eq. (A.5) and that
the induced bond market is arbitrage free. Then there exists ad-dimensional vector process
λ(t) = [λ1(t), ..., λd(t)] with the property that for allT ≥ 0 and for allt ≤ T, we have
67
α(t,T) = σ(t,T)∫ T
tσ′(t, s)ds− σ(t,T)λ(t), (A.8)
where ’ denotes transpose.
Schematically, the use of the HJM model can now be written as follows:
1. Specification of the volatilitiesσ(t,T)
2. The drift parameters of the forward rates are then uniquely determinedby Eq. (A.8).
3. Integrate the forward rate dynamics to get the forward rates as
f (t,T) = f ∗(t,T) +∫ t
0α(s,T)ds+
∫ t
0σ(s,T)dW(s). (A.9)
A.3 Risk Premium
Lets assume thatθ(t) is a 4 dimesional vector of real valued constants;θ(t) = (θ, θ(1), θ(2), θ(3))
Zθ(t) = exp
(∫ t
0θdB(u) − 1
2
∫ t
0θ2du
)(A.10)
and for i=1,2,3
Zθ(t) = exp
(∫ t
0θdJ(i)(u) − φ(i)(t, s,−iθ)
), (A.11)
whereφ(i) is the corresponding log-moment generating function.
Lets define an equivalent probability measureQθ such thatZθ(t) is the density process of
the Radon-Nikodym derivativedQθ/dP. Then with respect to probability measureQθ the
processes
Bθ(t) = B(t) − θt (A.12)
are Brownian motions. And the characteristic function ofJ(i) for i = 1,2,3, is
Eθ[eiuJ(i) |Ft] = exp(iu∫ t
0
∫
|z|<1z(eθz− 1)l(du,dz) +
∫ t
0
∫
R(eiuz− 1− iuzI|z|<1)eθzl(dz,du))
(A.13)
Therefore, according to the definition of the forward risk premium,
R(t, τ) := EQθ [S(τ)|Ft] − E[S(τ)|Ft], (A.14)
68
R(t, τ) = Λ(t) (Θ(t, τ, θ) − Θ(t, τ,0)) exp(12
∫ τ
tσ2(u)du) exp(
∫ τ
tσ(u)dB(u))
exp(∫ τ
tdJ(1)(u)) exp(e−νtY(t)) exp(e−βtZ(t)), (A.15)
where
lnΘ(t, τ, θ) =3∑
i=1
φ(i)(t, τ,e−d f(i)t + θ) − φ(i)(t, τ, θ) +∫ τ
tσ(u)θdu, (A.16)
d f(i) representing the corresponding discount factor, 0 fori = 1, ν for i = 2 andβ for i = 3.
69
APPENDIX B
Matlab Codes
In this appendix, Matlab codes used for jump detection and separation of jumpprocesses from
Brownian motion observations and classification of jumps are given. Thesethree procedures
are original contributions of this research. Therefore leaving commonly used seasonality
function fitting or Monte Carlo simulation procedures to the reader only these jump process
detection and classification codes are given here.
First of all as it is alraedy stated deterministic seasonality function is fitted to logarithmic price
series. Then residual series is separated to its stochastic components. This separation proce-
dure starts with iterative filterin of mean reverting jump processes. Corresponding matlab
codes used in this filterin is as follows.
function [Jumpsmr, Jumps, Filteredretf, Volf, GARCHP]=jumpfilter1(diffRes,cons)
%diffRes: first difference of deseasonalized logarithmic prices
%cons: mutiplier of estimated stochastic volatility, generally in the neighborhood of 3
Filteredret=zeros(length(diffRes),601);
Filteredret(:,1)=diffRes;
Vol=zeros(length(diffRes),600);
Retfiltered=zeros(length(diffRes),600);
for k=1:600
Jumps(:,k) Retfiltered(:,k) Vol(:,k) GARCHP(:,k)
=jump(Filteredret(:,k),cons);
Jumpsmr(:,1:k) Filteredret(:,k+1) Vol(:,k)
=jumpsmeanrev(Jumps(:,1:k),cons,diffRes,Vol(:,k),Retfiltered(:,k),GARCHP(:,k));
if Filteredret(:,k+1)-Filteredret(:,k)==zeros(length(Filteredret(:,k)),1);
Filteredretf=Filteredret(:,k+1);
70
Volf=Vol(:,k);
GARCHP=GARCHP(:,k);
break
end
end
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
function [Jumps, Retfiltered, Vol,GARCHP]=jump(diffRes,cons)
Coeff
=garchfit(diffRes);
Retfiltered=diffRes;
Jumps=zeros(length(diffRes),1);
c=Coeff.K;
a=Coeff.ARCH;
b=Coeff.GARCH;
Vol=zeros(length(diffRes),1);
Vol(1)=c+a*(mean(diffRes))2 + b ∗ (std(di f f Res))2;
f ori = 2 : length(di f f Res)
Vol(i) = c+ a ∗ (Ret f iltered(i − 1))2 + b ∗ Vol(i − 1);
i f (di f f Res(i))2 >= cons2 ∗ Vol(i);
Ret f iltered(i) = 0;
Jumps(i) = di f f Res(i);
break
end
end
Coeff
= garch f it(Ret f iltered);
c f = Coe f f.K;
a f = Coe f f.ARCH;
b f = Coe f f.GARCH;
GARCHP= [c f,a f,b f ]′;
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
f unction[Jumpsmr, Filteredret f,Vol] = jumpsmeanrev2(Jumps f, cons,di f f Res,Vol,
Filteredret f,GARCHP)
71
x = size(Jumps f);
row = x(1);
col = x(2);
c f = GARCHP(1);
a f = GARCHP(2);
b f = GARCHP(3);
all jumps= sum(Jumps f,2);
Jumps f1 = zeros(row, col);
rank1 = zeros(row,1);
f or j = 1 : col
f ori = 1 : row
i f Jumps f(i, j) = 0
rank1(i) = j;
break
end
end
end
nr = 0;
rank2 = zeros(row,1);
f ori = 1 : row;
i f all jumps(i) = 0;
nr = nr + 1;
rank2(i) = nr;
end
end
f ori = 1 : row
i f all jumps(i) = 0;
Jumps f1(:, rank2(i)) = Jumps f(:, rank1(i));
end
end
Jumpsmr= zeros(row, col);
f or j = 1 : col;
f ori = 1 : row− 7;
72
i f Jumps f1(i, j) = 0sum(Jumpsmr(i, :)) == 0;
Jumpsmr(i, j) = di f f Res(i);
i f min(−di f f Res(i)−cons∗sqrt(Vol(i+1)),−cons∗sqrt(Vol(i+1))) < di f f Res(i+1)di f f Res(i+
1) < max(−di f f Res(i) + cons∗ sqrt(Vol(i + 1)), cons∗ sqrt(Vol(i + 1)));
Jumpsmr(i + 1, j) = di f f Res(i + 1);
Filteredret f(i + 1) = 0;
Vol(i + 2) = c f + a f ∗ (Filteredret f(i + 1))2 + b f ∗ Vol(i + 1);
i f di f f Res(i) > 0sum(di f f Res(i : i + 1)) < 0+ cons∗ sqrt(Vol(i + 2))
break
elsei f di f f Res(i) < 0sum(di f f Res(i : i + 1)) > 0− cons∗ sqrt(Vol(i + 2))
break
elsei f
min([((di f f Res(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) − cons∗
sqrt(Vol(i + 1)))/di f f Res(i)) + 1) ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/
di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)∗di f f Res(i), ((di f f Res
(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) + cons∗ sqrt(Vol(i +
1)))/di f f Res(i)) + 1) ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗
(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)∗di f f Res(i)])−cons∗sqrt(Vol(i+
2)) < di f f Res(i+2)di f f Res(i+2) < max([((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))
(((di f f Res(i + 1)− cons∗ sqrt(Vol(i + 1)))/di f f Res(i))+ 1)∗ di f f Res(i), ((di f f Res(i + 1)+
cons∗sqrt(Vol(i+1)))/di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+
1)∗ di f f Res(i), ((di f f Res(i + 1)− cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1)+
cons∗ sqrt(Vol(i + 1)))/di f f Res(i))+ 1)∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i +
1)))/di f f Res(i))∗ (((di f f Res(i+1)−cons∗ sqrt(Vol(i+1)))/di f f Res(i))+1)∗di f f Res(i)])+
cons∗ sqrt(Vol(i + 2));
Jumpsmr(i + 2, j) = di f f Res(i + 2);
Filteredret f(i + 2) = 0;
Vol(i + 3) = c f + a f ∗ (Filteredret f(i + 2))2 + b f ∗ Vol(i + 2);
i f di f f Res(i) > 0sum(di f f Res(i : i + 2)) < 0+ cons∗ sqrt(Vol(i + 3))
break
elsei f di f f Res(i) < 0sum(di f f Res(i : i + 2)) > 0− cons∗ sqrt(Vol(i + 3))
break
elsei f
73
min([((di f f Res(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) − cons∗
sqrt(Vol(i + 1)))/di f f Res(i)) + 1)2 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/
di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)2∗di f f Res(i), ((di f f Res
(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) + cons∗ sqrt(Vol(i +
1)))/di f f Res(i))+ 1)2 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗
(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)2∗di f f Res(i)])−cons∗sqrt(Vol(i+
3)) < di f f Res(i+3)di f f Res(i+3) < max([((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))
(((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)2 ∗di f f Res(i), ((di f f Res(i +1)+
cons∗sqrt(Vol(i+1)))/di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+
1)2 ∗di f f Res(i), ((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i)) ∗ (((di f f Res(i +1)+
cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)2 ∗di f f Res(i), ((di f f Res(i +1)+ cons∗ sqrt(Vol(i +
1)))/di f f Res(i))∗(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)2∗di f f Res(i)])+
cons∗ sqrt(Vol(i + 3));
Jumpsmr(i + 3, j) = di f f Res(i + 3);
Filteredret f(i + 3) = 0;
Vol(i + 4) = c f + a f ∗ (Filteredret f(i + 3))2 + b f ∗ Vol(i + 3);
i f di f f Res(i) > 0sum(di f f Res(i : i + 3)) < 0+ cons∗ sqrt(Vol(i + 4))
break
elsei f di f f Res(i) < 0sum(di f f Res(i : i + 3)) > 0− cons∗ sqrt(Vol(i + 4))
break
elsei f
min([((di f f Res(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) − cons∗
sqrt(Vol(i + 1)))/di f f Res(i)) + 1)3 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/
di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)3∗di f f Res(i), ((di f f Res
(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) + cons∗ sqrt(Vol(i +
1)))/di f f Res(i))+ 1)3 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗
(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)3∗di f f Res(i)])−cons∗sqrt(Vol(i+
4)) < di f f Res(i+4)...di f f Res(i+4) < max([((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))
(((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)3 ∗di f f Res(i), ((di f f Res(i +1)+
cons∗sqrt(Vol(i+1)))/di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+
1)3 ∗di f f Res(i), ((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i)) ∗ (((di f f Res(i +1)+
cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)3 ∗di f f Res(i), ((di f f Res(i +1)+ cons∗ sqrt(Vol(i +
1)))/di f f Res(i))∗(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)3∗di f f Res(i)])+
74
cons∗ sqrt(Vol(i + 4));
Jumpsmr(i + 4, j) = di f f Res(i + 4);
Filteredret f(i + 4) = 0;
Vol(i + 5) = c f + a f ∗ (Filteredret f(i + 4))2 + b f ∗ Vol(i + 4);
i f di f f Res(i) > 0sum(di f f Res(i : i + 4)) < 0+ cons∗ sqrt(Vol(i + 5))
break
elsei f di f f Res(i) < 0sum(di f f Res(i : i + 4)) > 0− cons∗ sqrt(Vol(i + 5))
break
elsei f
min([((di f f Res(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) − cons∗
sqrt(Vol(i + 1)))/di f f Res(i)) + 1)4 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/
di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)4∗di f f Res(i), ((di f f Res
(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) + cons∗ sqrt(Vol(i +
1)))/di f f Res(i))+ 1)4 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗
(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)4∗di f f Res(i)])−cons∗sqrt(Vol(i+
5)) < di f f Res(i+5)...di f f Res(i+5) < max([((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))
(((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)4 ∗di f f Res(i), ((di f f Res(i +1)+
cons∗sqrt(Vol(i+1)))/di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+
1)4 ∗di f f Res(i), ((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i)) ∗ (((di f f Res(i +1)+
cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)4 ∗di f f Res(i), ((di f f Res(i +1)+ cons∗ sqrt(Vol(i +
1)))/di f f Res(i))∗(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)4∗di f f Res(i)])+
cons∗ sqrt(Vol(i + 5));
Jumpsmr(i + 5, j) = di f f Res(i + 5);
Filteredret f(i + 5) = 0;
Vol(i + 6) = c f + a f ∗ (Filteredret f(i + 5))2 + b f ∗ Vol(i + 5);
i f di f f Res(i) > 0sum(di f f Res(i : i + 5)) < 0+ cons∗ sqrt(Vol(i + 6))
break
elsei f di f f Res(i) < 0sum(di f f Res(i : i + 5)) > 0− cons∗ sqrt(Vol(i + 6))
break
elsei f
min([((di f f Res(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) − cons∗
sqrt(Vol(i + 1)))/di f f Res(i)) + 1)5 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/
di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)5∗di f f Res(i), ((di f f Res
75
(i + 1) − cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗ (((di f f Res(i + 1) + cons∗ sqrt(Vol(i +
1)))/di f f Res(i))+ 1)5 ∗ di f f Res(i), ((di f f Res(i + 1)+ cons∗ sqrt(Vol(i + 1)))/di f f Res(i)) ∗
(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)5∗di f f Res(i)])−cons∗sqrt(Vol(i+
6)) < di f f Res(i+6)...di f f Res(i+6) < max([((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))
(((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)5 ∗di f f Res(i), ((di f f Res(i +1)+
cons∗sqrt(Vol(i+1)))/di f f Res(i))∗(((di f f Res(i+1)+cons∗sqrt(Vol(i+1)))/di f f Res(i))+
1)5 ∗di f f Res(i), ((di f f Res(i +1)− cons∗ sqrt(Vol(i +1)))/di f f Res(i)) ∗ (((di f f Res(i +1)+
cons∗ sqrt(Vol(i +1)))/di f f Res(i))+1)5 ∗di f f Res(i), ((di f f Res(i +1)+ cons∗ sqrt(Vol(i +
1)))/di f f Res(i))∗(((di f f Res(i+1)−cons∗sqrt(Vol(i+1)))/di f f Res(i))+1)5∗di f f Res(i)])+
cons∗ sqrt(Vol(i + 6));
Jumpsmr(i + 6, j) = di f f Res(i + 6);
Filteredret f(i + 6) = 0;
Vol(i + 7) = c f + a f ∗ (Filteredret f(i + 6))2 + b f ∗ Vol(i + 6);
break
else
break
end
else
break
end
else
break
end
else
break
end
else
break
end
else
break
end
end
76
end
f ori = 2 : row− 7
i f Jumpsmr(i, j) = 0Jumpsmr(i − 1, j) == 0;
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 6, j)))
Jumpsmr(i + 6, j) = 0;
Filteredret f(i + 6) = di f f Res(i + 6);
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 5, j)))
Jumpsmr(i + 5, j) = 0;
Filteredret f(i + 5) = di f f Res(i + 5);
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 4, j)))
Jumpsmr(i + 4, j) = 0;
Filteredret f(i + 4) = di f f Res(i + 4);
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 3, j)))
Jumpsmr(i + 3, j) = 0;
Filteredret f(i + 3) = di f f Res(i + 3);
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 2, j)))
Jumpsmr(i + 2, j) = 0;
Filteredret f(i + 2) = di f f Res(i + 2);
i f abs(di f f Res(i)) < abs(sum(Jumpsmr(i : i + 1, j)))
Jumpsmr(i + 1, j) = 0;
Filteredret f(i + 1) = di f f Res(i + 1);
end
end
end
end
end
end
end
end
f ori = 2 : row− 7
i f Jumpsmr(i, j) = 0Jumpsmr(i − 1, j) == 0;
i f 0.06∗ abs(di f f Res(i)) > abs(sum(Jumpsmr(i + 1 : i + 6, j)))
Jumpsmr(i + 1, j) = 0;
77
Filteredret f(i + 1) = di f f Res(i + 1);
Jumpsmr(i + 2, j) = 0;
Filteredret f(i + 2) = di f f Res(i + 2);
Jumpsmr(i + 3, j) = 0;
Filteredret f(i + 3) = di f f Res(i + 3);
Jumpsmr(i + 4, j) = 0;
Filteredret f(i + 4) = di f f Res(i + 4);
Jumpsmr(i + 5, j) = 0;
Filteredret f(i + 5) = di f f Res(i + 5);
Jumpsmr(i + 6, j) = 0;
Filteredret f(i + 6) = di f f Res(i + 6);
end
end
end
Vol(i + 2) = c f + a f ∗ (Filteredret f(i + 1))2 + b f ∗ Vol(i + 1);
Vol(i + 3) = c f + a f ∗ (Filteredret f(i + 2))2 + b f ∗ Vol(i + 2);
Vol(i + 4) = c f + a f ∗ (Filteredret f(i + 3))2 + b f ∗ Vol(i + 3);
Vol(i + 5) = c f + a f ∗ (Filteredret f(i + 4))2 + b f ∗ Vol(i + 4);
Vol(i + 6) = c f + a f ∗ (Filteredret f(i + 5))2 + b f ∗ Vol(i + 5);
Vol(i + 7) = c f + a f ∗ (Filteredret f(i + 6))2 + b f ∗ Vol(i + 6);
end
Filteredret f = di f f Res− sum(Jumpsmr,2);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Classi f ication o f the detected jump points.
function [pj,sp,ss, noj, sp0, ss0]=jumpclass(diffRes, Jumpsmr, Vol,cons)
%pj: vector of pure jump points
%sp: vector of detected spike processes
%ss: vector of detected semi spike processes
%noj: number of jumps
%sp0: only the spike jump points
%ss0: only the semi-spike jump points
x=size(Jumpsmr);
78
row=x(1);
col=x(2);
pj=zeros(row,1);
sp=zeros(row,1);
ss=zeros(row,1);
sp0=zeros(row,1);
ss0=zeros(row,1);
noj=zeros(1,5);
for j=1:col
for i=2:row
if Jumpsmr(i,j)=0 Jumpsmr(i+1,j)==0 Jumpsmr(i-1,j)==0;
pj(i)=diffRes(i);
break
elseif Jumpsmr(i,j)=0 Jumpsmr(i+1,j)=0 sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+1,j)) sign(Jumpsmr(i-
1,j))==0;
if abs(Jumpsmr(i,j))-cons*sqrt(Vol(i+1))¡=abs(Jumpsmr(i+1,j));
if Jumpsmr(i+2,j)==0
sp(i)=diffRes(i);
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
break
elseif sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+2,j)) Jumpsmr(i+2,j)=0 abs(sum(Jumpsmr(i:i+1,j)))-
cons*sqrt(Vol(i+2))¡=abs(Jumpsmr(i+2,j));
if Jumpsmr(i+3,j)==0
sp(i)=diffRes(i);
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
sp(i+2)=diffRes(i+2);
break
elseif sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+3,j)) Jumpsmr(i+3,j)=0 abs(sum(Jumpsmr(i:i+2,j)))-
cons*sqrt(Vol(i+3))¡=abs(Jumpsmr(i+3,j));
if Jumpsmr(i+4,j)==0
sp(i)=diffRes(i);
79
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
sp(i+2)=diffRes(i+2);
sp(i+3)=diffRes(i+3);
break
elseif sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+4,j)) Jumpsmr(i+4,j) =0
abs(sum(Jumpsmr(i:i+3,j)))-cons*sqrt(Vol(i+4))¡=abs(Jumpsmr(i+4,j));
if Jumpsmr(i+5,j)==0
sp(i)=diffRes(i);
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
sp(i+2)=diffRes(i+2);
sp(i+3)=diffRes(i+3);
sp(i+4)=diffRes(i+4);
break
elseif sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+5,j)) Jumpsmr(i+5,j)=0 abs(sum(Jumpsmr(i:i+4,j)))-
cons*sqrt(Vol(i+5))¡=abs(Jumpsmr(i+5,j));
if Jumpsmr(i+6,j)==0
sp(i)=diffRes(i);
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
sp(i+2)=diffRes(i+2);
sp(i+3)=diffRes(i+3);
sp(i+4)=diffRes(i+4);
sp(i+5)=diffRes(i+5);
break
elseif sign(Jumpsmr(i,j))==-sign(Jumpsmr(i+6,j)) Jumpsmr(i+6,j)=0 abs(sum(Jumpsmr(i:i+5,j)))-
cons*sqrt(Vol(i+6))¡=abs(Jumpsmr(i+6,j));
sp(i)=diffRes(i);
sp0(i)=diffRes(i);
sp(i+1)=diffRes(i+1);
sp(i+2)=diffRes(i+2);
sp(i+3)=diffRes(i+3);
80
sp(i+4)=diffRes(i+4);
sp(i+5)=diffRes(i+5);
sp(i+6)=diffRes(i+6);
break
end
end
end
end
end
end
end
end
end
ss=sum(Jumpsmr,2)-pj-sp;
for i=1:row
if ss(i) =0 ss(i-1)==0
ss0(i)=diffRes(i);
end
end
for i=1:row
if pj(i) =0
noj(1)=noj(1)+1;
end
end
for i=1:row
if sp0(i) =0
noj(2)=noj(2)+1;
end
end
for i=1:row
if ss0(i)=0
noj(3)=noj(3)+1;
end
81
end
for i=1:row
if sp(i) =0
noj(4)=noj(4)+1;
end
end
for i=1:row
if ss(i) =0
noj(5)=noj(5)+1;
end
end
82
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VITA
PERSONAL INFORMATION
Surname, Name: Talaslı,Irem
Nationality: Turkish (TC)
Date and Place of Birth: 15 August 1981, Ankara
email: [email protected]
EDUCATION
Ph.D. Department of Financial Mathematics, 2012, January
Institute of Applied Mathematics
Middle East Technical University, Ankara
Supervisor: Assoc. Prof. Dr. Azize Hayfavi
Thesis Title: Stochastic Modeling of Electricity Markets
M.Sc. Department of Financial Mathematics, 2005, July
Institute of Applied Mathematics
Middle East Technical University, Ankara
Supervisor: Prof. Dr. Hayri Korezlioglu
Thesis Title: Coherent Risk Measures
B.Sc. Department of Economics, 2003, June
Middle East Technical University, Ankara
WORK EXPERIENCE TRAINING
July 2006-PresentCentral Bank of the Republic of Turkey, Markets Division, Ankara
Feb - April 2006 PricewaterhouseCoopers, Financial Risk Dept,Istanbul - Intern
Aug 2004Istanbul Stock Exchange, Research Dept.,Istanbul - Intern
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